aa r X i v : . [ m a t h . GN ] O c t SELECTIONS AND HIGHER SEPARATION AXIOMS
VALENTIN GUTEV
Abstract.
This survey presents some historical background and recent devel-opments in the area of selections for set-valued mappings along with severalopen questions. It was written with the hope that the presented material maypique an interest in the selection problem for set-valued mappings — a problemwith a fascinating history and appealing applications.
1. The Selection Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Selections and Dowker’s Extension Theorem . . . . . . . . . . . . . . . . . 2 ⋄ ⋄ ⋄ ⋄ ⋄
3. Selections and Continuity-Like Properties . . . . . . . . . . . . . . . . . . . 11 ⋄ ⋄ ⋄ ⋄
4. Selections and Compact-Like Families. . . . . . . . . . . . . . . . . . . . . . . 17 ⋄ ⋄ ⋄ ⋄ ⋄ ℓ p ( A )-spaces (23).
5. Selections and Finite-Dimensional Spaces. . . . . . . . . . . . . . . . . . . . 24 ⋄ ⋄ ⋄ Date : October 16, 2018.2010
Mathematics Subject Classification.
Primary 54C60, 54C65; Secondary 46A25, 46B10,46C05, 54C20, 54C55, 54D15, 54F35, 54F45, 55P05.
Key words and phrases.
Set-valued mapping, lower semi-continuous, continuous selection,continuous extension, reflexive Banach space, finite-dimensional space, infinite-dimensionalspace.
6. Selections and Infinite-Dimensional Spaces. . . . . . . . . . . . . . . . . . . 29 ⋄ C -spaces (30) ⋄ ⋄ G δ -Sets (33) ⋄
1. The Selection Problem
In the mid 1950’s Ernest Michael wrote a series of fundamental papers [54,55, 56] relating familiar extension theorems to selections, thus laying down thefoundation of the theory of continuous selections. Nowadays, selections becamean indispensable tool for many mathematicians working in vastly different areas.However, the key importance of Michael’s selection theory is not only in providinga comprehensive solution to diverse selection problems, but also in the immedi-ate inclusion of the obtained results into the general context of development oftopology. In the first of these papers [54], Michael wrote:“One of the most interesting and important problems in topology is the exten-sion problem : Two topological spaces X and Y are given, together with a closed A ⊂ X , and we would like to know whether every continuous function g : A → Y can be extended to a continuous function f from X (or at least from some open U ⊃ A ) into Y . Sometimes there are additional requirements on f , which fre-quently (as in the theory of fibre bundles) take the following form: For every x in X , f ( x ) must be an element of a pre-assigned subset of Y . This new problem,which we call the selection problem , is clearly more general than the extensionproblem, and presents a challenge even when A is the null set or a 1-point set(where the extension problem is trivial).”Michael proceeded to remark that with the exception of sandwich-like theorems,such as Tong [91, 92], Katˇetov [50, 51] and Dowker [16], no attempt has beenmade to obtain results under minimal hypotheses. Then he stated the purposeof his papers emphasising that most of the familiar extension theorems, such asthe Tietze-Urysohn extension theorem, Kuratowski’s extension theorem for finite-dimensional spaces and the Borsuk homotopy extension theorem can be slightlyaltered and essentially generalised in terms of selection theorems.The present survey contains different aspects of Michael’s selection theory, pos-sible new research directions along with several open questions.
2. Selections and Dowker’s Extension Theorem
In this section, and the rest of the paper, all spaces are Hausdorff topologicalspaces. However, several of the considerations are also valid for T -spaces, andsome even without assuming any separation axioms. ELECTIONS AND HIGHER SEPARATION AXIOMS 3
A space X is collectionwise normal (Bing [7])if for every discrete collection D of subsets of X there exists a discrete collection { U D : D ∈ D } of open subsets of X such that D ⊂ U D , for all D ∈ D . In thiscase, we simply say that D has an open discrete expansion . Every collectionwisenormal space is normal, but the converse is not necessarily true. In the samepaper, see [7, Examples G and H], Bing described an example of a normal spacewhich is not collectionwise normal, now known as Bing’s example [24, 5.1.23Bing’s Example]. He also proved that full normality (i.e. paracompactness) impliescollectionwise normality but not conversely [7, Theorem 12]. Recall that a space X is paracompact if every open cover of X has a locally finite open refinement.Since the closure of the elements of any discrete collection is a discrete collectiontoo, a space X is collectionwise normal if every discrete collection D of closedsubsets of X has an open discrete expansion. In the presence of normality, this canbe further relaxed to requiring D to have only an open pairwise-disjoint expansion (i.e. an open pairwise-disjoint family { U D : D ∈ D } with D ⊂ U D , for all D ∈ D ).Collectionwise normality is a natural generalisation of normality. Indeed, if X has this property with respect to discrete families D of cardinality at most τ , thenit is called τ -collectionwise normal . Thus, X is normal iff it is 2-collectionwisenormal (or, more generally, n -collectionwise normal for every finite n ≥ X is normal iff it is ω -collectionwise normal, which follows easily fromUrysohn’s characterisation of normality. On the other hand, for every infinite car-dinal τ there exists a τ -collectionwise normal space which is not τ + -collectionwisenormal [74], where the cardinal τ + is the immediate successor of τ .Collectionwise normality is the natural domain for continuous extensions. Thefollowing extension theorem was proved by Dowker [17], and is commonly called Dowker’s extension theorem . Theorem 2.1 ([17]) . A space X is collectionwise normal if and only if for everyclosed subset A ⊂ X , every continuous map from A to a Banach space E can becontinuously extended to the whole of X . For a simple proof of Theorem 2.1, the interested reader is referred to [38].
For a space Y , let F ( Y ) be the collection ofall nonempty closed subsets of Y ; C ( Y ) — that of all compact members of F ( Y );and C ′ ( Y ) = C ( Y ) ∪ { Y } . For a (not necessarily convex) subset Y of a normedspace E , we will use the subscript “ c ” to denote the convex members of anyoneof the above families; namely, F c ( Y ) for the convex members of F ( Y ); C c ( Y ) forthose of C ( Y ); and C ′ c ( Y ) = C c ( Y ) ∪ { Y } .For spaces (sets) X and Y , we write Φ : X Y to designate that Φ is a set-valued (or multi-valued ) mapping from X to the nonempty subsets of Y . Such a VALENTIN GUTEV mapping Φ : X Y is lower semi-continuous , or l.s.c. , if the setΦ − [ U ] = { x ∈ X : Φ( x ) ∩ U = ∅ } is open in X , for every open U ⊂ Y . Finally, let us recall that a map f : X → Y is a selection (or a single-valued selection ) for Φ : X Y if f ( x ) ∈ Φ( x ), for every x ∈ X . The following natural relationship between selections and extensions iswell known. Proposition 2.2.
For a subset A ⊂ X and g : A → Y , define a set-valuedmapping Φ g : X → C ′ ( Y ) by Φ g ( x ) = { g ( x ) } if x ∈ A , and Φ g ( x ) = Y otherwise.Then (i) f : X → Y is a continuous selection for Φ g if and only if it is a continuousextension for g . (ii) If Y has at least two points, then Φ g is l.s.c. if and only if A is closed and g is continuous. Proposition 2.2(i) is [54, Example 1.3], and follows immediately from the def-inition of Φ g . The one direction of Proposition 2.2(ii) is [54, Example 1.3*]; theother follows from the fact that Y contains nonempty disjoint open sets U and V ,hence X \ A = Φ − g [ U ] ∩ Φ − g [ V ].According to Proposition 2.2, any extension theorem can be restated as a se-lection theorem for the mapping Φ g . Based on this, Michael [54, 55, 56] proposedseveral selection theorems generalising ordinary extension theorems. Here are twoof them whose prototype is Dowker’s extension theorem (Theorem 2.1). Theorem 2.3 ([54]) . A space X is collectionwise normal if and only if for everyBanach space E , every l.s.c. mapping Φ : X → C ′ c ( E ) has a continuous selection. Theorem 2.4 ([54]) . A space X is paracompact if and only if for every Banachspace E , every l.s.c. mapping Φ : X → F c ( E ) has a continuous selection. However, the proof of Theorem 2.3 in [54] was incomplete and, in fact, workingonly for the case of C c ( E )-valued mappings. The first complete proof of thistheorem was given by Choban and Valov [15] using a different technique. Asimple proof of Theorem 2.3, based on Theorem 2.1, was given in [39]. For an infinite cardinal number τ , a space X is called τ -PF-normal (see [89]) if every point-finite open cover of X of cardinality ≤ τ isnormal. Every τ -collectionwise normal space is τ -PF-normal (see [53]), and ω -PF-normality coincides with normality [63]. However, PF-normality is neitheridentical to collectionwise normality (see Bing’s example [7] and [53, Example 1]),nor to normality ([53, Example 2]). For some properties of PF-normal spaces, theinterested reader is referred to [41, Section 3] and [53]. ELECTIONS AND HIGHER SEPARATION AXIOMS 5
In the realm of normal spaces, PF-normal spaces coincide with the point-finitelyparacompact spaces in the sense of Kandˆo [47]; while, in Nedev’s terminology [69], τ -PF-normal spaces are precisely the τ -pointwise- ℵ -paracompact spaces . Thefollowing characterisation of PF-normal spaces was obtained by Kandˆo [47] andNedev [69]. Theorem 2.5 ([47, 69]) . A space X is PF-normal if and only if for every normedspace E , every l.s.c. mapping ϕ : X → C c ( E ) has a continuous selection. For a metric space (
Y, ρ ) and ε >
0, we will use O ε ( p ) for the open ε -ball centred at a point p ∈ Y ; and O ε ( S ) = S p ∈ S O ε ( p ), whenever S ⊂ Y . A map f : X → Y is an ε -selection for amapping Φ : X Y if f ( x ) ∈ O ε (Φ( x )), for every x ∈ X .Although Theorem 2.3 is valid in the way it was stated by Michael, it was shownin [39, Theorem 1.2] that it is equivalent to both Theorem 2.1 and Theorem 2.5.In fact, a bit more was obtained in [39], see [39, Claim 2.1]. Theorem 2.6 ([39]) . Let E be a Banach space. Then for a space X , the followingare equivalent :(a) Every l.s.c. mapping
Φ : X → C ′ c ( E ) has a continuous selection. (b) If A ⊂ X is closed, then every l.s.c. mapping ϕ : A → C c ( E ) has acontinuous selection, and every continuous g : A → E can be extended toa continuous map f : X → E . (c) If Φ : X E is an l.s.c. convex-valued mapping which admits a continuousmap g : X → E such that Φ( x ) is compact whenever g ( x ) / ∈ Φ( x ) , then Φ has a continuous ε -selection, for every ε > . It should be remarked that the proof of Theorem 2.6 is straightforward avoid-ing any explicit reference to collectionwise normality. It should be also remarkedthat the equivalence (a) ⇔ (b) in Theorem 2.6 represents the selection propertyof PF-normality (Theorem 2.5) which, together with Dowker’s extension theorem(Theorem 2.1), gives Theorem 2.3. This brings the natural question of whetherthere is any selection generalisation of Dowker’s extension theorem. Various as-pects of this question are discussed below. Motivated by (c) of Theorem 2.6, thefollowing question was posed in [39, Question 1].
Question 1 ([39]) . Let X be a collectionwise normal space and E be a Banachspace. Suppose that Φ : X → F c ( E ) is an l.s.c. mapping which admits a contin-uous map g : X → E with Φ( x ) compact, whenever g ( x ) / ∈ Φ( x ) for some x ∈ X .Does Φ have a continuous selection?If Φ : X → C ′ ( E ) and g : X → E is any map, then Φ( x ) is compact forevery x ∈ X with g ( x ) / ∈ Φ( x ). Moreover, for a collectionwise normal space X VALENTIN GUTEV and a Banach space E , every l.s.c. mapping Φ : X → C ′ c ( E ) has a continuousselection, by Theorem 2.3. Thus, the answer to Question 1 is “Yes” for C ′ c ( E )-valued mappings. A natural example of mappings which are as in Question 1, butnot C ′ c ( E )-valued, is given at the end of this section, see Remark 2.20.If ∈ E is the origin of a normed space E and ϕ : X → C ′ ( E ), then ϕ ( x ) iscompact for every x ∈ X with / ∈ ϕ ( x ). This property is equivalent to the onein Question 1. Indeed, let Φ : X → F c ( E ) and g : X → E be a continuous mapwith Φ( x ) compact, whenever g ( x ) / ∈ Φ( x ) for some x ∈ X . Define a mapping ϕ : X → F c ( E ) by ϕ ( x ) = Φ( x ) − g ( x ), x ∈ X . Then ϕ ( x ) is compact for every x ∈ X with / ∈ ϕ ( x ), and ϕ has a continuous selection if and only if so does Φ.Thus, we have the following relaxed form of Question 1. Question 2.
Let X be a collectionwise normal space, E be a Banach space and ϕ : X → F c ( E ) be an l.s.c. mapping such that ϕ ( x ) is compact, for every x ∈ X with / ∈ ϕ ( x ). Does ϕ have a continuous selection?Here are some further remarks regarding some of the challenges in this question. Proposition 2.7.
Let E be a metrizable space, ϕ : X → F ( E ) be l.s.c. and g : X → E be continuous. Then { x ∈ X : g ( x ) ∈ ϕ ( x ) } is a G δ -set in X . Complementary to Proposition 2.7 is the following observation.
Proposition 2.8.
Let X be a collectionwise normal space, E be a normed spaceand ϕ : X → F c ( E ) be an l.s.c. mapping such that ϕ ( x ) is compact, for every x ∈ X with / ∈ ϕ ( x ) . Set H = { x ∈ X : / ∈ ϕ ( x ) } . Then ϕ ↾ H has a continuousselection. If, moreover, E is a Banach space, then ϕ ↾ G has a continuous selectionfor some G δ -subset G ⊂ X with H ⊂ G .Proof. According to [80, Theorem 1.3] and Proposition 2.7, H is itself a collec-tionwise normal space. Hence, by Theorem 2.3 (see also Theorem 2.5), ϕ ↾ H has a continuous selection h : H → E . If E is a Banach space, then h can beextended to a continuous map g : Z → E for some G δ -subset Z ⊂ X contain-ing H , see [24, Theorem 4.3.20]. Applying Proposition 2.7 once more, the set G = { x ∈ Z : g ( x ) ∈ ϕ ( x ) } is also G δ , and clearly contains H . (cid:3) Propositions 2.7 and 2.8 show that the mapping ϕ : X → F c ( E ) in Question2 has two partial continuous selections on complementary subsets of the domain.Hence, a particular challenge in this question (and in Question 1 as well) is if onecan use these partial selections to construct a continuous selection for the mapping ϕ itself. This brings the following alternative question. Question 3.
Let X be a collectionwise normal space, E be a Banach space, ϕ : X → F c ( E ) be an l.s.c. mapping and H = { x ∈ X : / ∈ ϕ ( x ) } . Does thereexist a continuous selection for ϕ provided ϕ ↾ H has a continuous selection? ELECTIONS AND HIGHER SEPARATION AXIOMS 7
The property in Question 3 implies collectionwise normality of X . Indeed, takea closed set A ⊂ X and a continuous map g : A → E into a Banach space E .Next, let Φ g : X → C ′ c ( E ) be the associated mapping defined as in Proposition2.2. Then Φ g is l.s.c. and H = { x ∈ X : / ∈ Φ g ( x ) } ⊂ A . Hence, g ↾ H isa continuous selection for Φ g ↾ H . Thus, by Proposition 2.2, the existence of acontinuous selection for Φ g (as per Question 3) is equivalent to the existence ofa continuous extension of g , i.e. to collectionwise normality of X (by Theorem2.1). This suggests the following interpretation of Question 3 in the setting ofPF-normal spaces. Question 4.
Let ϕ : X → F c ( E ) be an l.s.c. mapping, where X is a PF-normalspace and E is a Banach space. Suppose that there exists a continuous map h : X → E such that h ( x ) ∈ ϕ ( x ), for every x ∈ X with / ∈ ϕ ( x ). Does ϕ have acontinuous selection? In what follows, to each Φ : X Y we willassociate the mapping Φ : X → F ( Y ) defined by Φ( x ) = Φ( x ), x ∈ X . Moreover,for a pair of mappings Φ , Ψ : X Y with Φ( x ) ∩ Ψ( x ) = ∅ , x ∈ X , we will useΦ ∧ Ψ to denote their intersection, i.e. the mapping which assigns to each x ∈ X the set [Φ ∧ Ψ]( x ) = Φ( x ) ∩ Ψ( x ). The graph of a mapping Ψ : X Y is theset { ( x, y ) ∈ X × Y : y ∈ Ψ( x ) } , and we say that Ψ has an open ( closed ) graphif its graph is open (respectively, closed) in X × Y . Finally, to each ε > X Y into a metric space ( Y, ρ ), we will associate the mapping O ε [Φ] : X Y defined by O ε [Φ]( x ) = O ε (Φ( x )), x ∈ X . This convention will bealso used in an obvious manner for usual maps f : X → Y considering f as thesingleton-valued mapping x → { f ( x ) } , x ∈ X . In these terms, for f, g : X → Y and ε, µ >
0, we have that f is a µ -selection for Φ : X Y with ρ ( f ( x ) , g ( x )) < ε for every x ∈ X , if and only if f is a selection for the mapping O µ [Φ] ∧ O ε [ g ].The following three observations are due to Michael, see [54, Propositions 2.3and 2.5] and [55, Lemma 11.3]. For a paracompact domain, they reduce theselection problem for l.s.c. mappings to that one of approximate selections (seeProposition 2.12). Proposition 2.9 ([54]) . For spaces X and Y , a mapping Φ : X Y is l.s.c. ifand only if so is the mapping Φ : X → F ( Y ) . Proposition 2.10 ([54]) . Let
Φ : X Y be an l.s.c. mapping and Ψ : X Y bean open-graph mapping with Φ( x ) ∩ Ψ( x ) = ∅ , for all x ∈ X . Then the mapping Φ ∧ Ψ : X Y is also l.s.c. Proposition 2.11 ([55]) . If ( Y, ρ ) is a metric space, ε > and Φ : X Y is l.s.c.,then the associated mapping O ε [Φ] : X Y has an open graph. In particular, foreach compact set K ⊂ Y , the set { x ∈ X : K ⊂ O ε [Φ]( x ) } is open in X . VALENTIN GUTEV
We now have the following general observation which is a partial case of [55,Proof that Lemma 5.1 implies Theorem 4.1, page 569].
Proposition 2.12 ([55]) . Let E be a Banach space. Then for a space X , thefollowing are equivalent :(a) Each l.s.c. mapping
Φ : X → F c ( E ) has a continuous selection. (b) Each l.s.c. convex-valued mapping
Φ : X E has a continuous ε -selection,for every ε > .Proof. If Φ : X E is an l.s.c. convex-valued mapping, then by Proposition 2.9,so is Φ : X → F c ( E ). Moreover, if f : X → E is a selection for Φ, then f isan ε -selection for Φ, for every ε >
0. This shows that (a) ⇒ (b). To see that(b) ⇒ (a), suppose that (b) holds, and Φ : X → F c ( E ) is l.s.c. Then by (b), Φhas a continuous 2 − -selection f : X → E . According to Propositions 2.10 and2.11, Φ ∧ O − [ f ] : X E is also l.s.c., and clearly it is convex-valued. Hence,for the same reason, it has a continuous 2 − -selection f : X → E . Evidently, f is a continuous 2 − -selection for Φ with k f ( x ) − f ( x ) k < − + 2 − < , x ∈ X .The construction can be carried on by induction to get a sequence f n : X → E , n ∈ N , of continuous 2 − n -selections for Φ such that k f n +1 ( x ) − f n ( x ) k < − n +1 , n ∈ N . This resulting sequence of maps is uniformly Cauchy, hence it convergesto some continuous map f : X → E because E is a Banach space; and this f is acontinuous selection for Φ as required in (a). (cid:3) Complementary to Proposition 2.12 is the following consequence of Theorems2.3 and 2.6.
Corollary 2.13.
Let E be a Banach space, X be a collectionwise normal spaceand Φ : X E be an l.s.c. convex-valued mapping such that Φ( x ) is compact, forevery x ∈ X with / ∈ Φ( x ) . Then Φ has a continuous ε -selection, for every ε > . This brings the following alternative question, which represents another aspectof Question 2 (hence, of Question 1 as well).
Question 5.
Let X be a collectionwise normal space, E be a Banach space andΦ : X → F c ( E ) be an l.s.c. mapping which has a continuous ε -selection, for every ε >
0. Does Φ have a continuous selection?Evidently, the answer is “Yes” if Φ is a C ′ c ( E )-valued mapping. Moreover, theproperty in Question 5 implies collectionwise normality. Proposition 2.14.
Let X be a space such that for every Banach space E , everyl.s.c. mapping Φ : X → C ′ c ( E ) has a continuous ε -selection, for every ε > . Then X is collectionwise normal.Proof. Let A ⊂ X be closed, E be a Banach space and g : A → E be a continuousmap. Then the mapping Φ g : X → C ′ c ( E ), defined as in Proposition 2.2, is l.s.c. ELECTIONS AND HIGHER SEPARATION AXIOMS 9
Hence, by condition, Φ g has a continuous ε -selection, for every ε >
0. In otherwords, for every ε >
0, the mapping Φ( x ) = E , x ∈ X , has a continuous selection h : X → E such that k h ( x ) − g ( x ) k < ε , for all x ∈ A . According to [42, Lemma4.2], g can be extended to a continuous map f : X → E . Hence, by Theorem 2.1, X is collectionwise normal. (cid:3) If X is only assumed to be PF-normal, then the answer to Question 5 is “Yes”if each Φ( x ) is either compact or finite-dimensional. The latter means that Φ( x )is contained in some finite-dimensional affine subspace of E ; equivalently, thatΦ( x ) ⊂ q + L for some q ∈ E and a finite-dimensional linear subspace L ⊂ E . Proposition 2.15.
Let X be a PF-normal space, E be a normed space and Φ : X → F c ( E ) be an l.s.c. mapping such that each Φ( x ) is either compact orfinite-dimensional. If Φ has a continuous ε -selection for some ε > , then it alsohas a continuous selection.Proof. Let g : X → E be a continuous ε -selection for Φ, for some ε >
0. Then byby Propositions 2.10 and 2.11, the intersection mapping Φ ∧ O ε [ g ] : X E is alsol.s.c. Hence, by Proposition 2.9, so is the mapping ϕ = Φ ∧ O ε [ g ] : X → F c ( E ).Furthermore, ϕ is compact-valued. Indeed, if Φ( x ) is compact, then so is ϕ ( x ). IfΦ( x ) is finite-dimensional, then Φ( x ) ⊂ q + L for some finite-dimensional linearsubspace L ⊂ E and q ∈ E . However, L is complete with respect to the normof E (being finite-dimensional) and ϕ ( x ) − q ⊂ L is closed and bounded in L .Therefore, ϕ ( x ) is compact being a translate of the compact set ϕ ( x ) − q . Thus,by Theorem 2.5, ϕ has a continuous selection f : X → E . This f is also a selectionfor Φ because ϕ ( x ) ⊂ Φ( x ), for every x ∈ X . (cid:3) For an infinite cardinal number τ , a space X is called τ -paracompact if everyopen cover of X of cardinality ≤ τ has a locally finite open refinement. The ω -paracompact spaces are commonly called countably paracompact . A mapping ϕ : X E is a set-valued selection (or set-selection , or multi-selection ) for amapping Φ : X E if ϕ ( x ) ⊂ Φ( x ), for all x ∈ X . A mapping ϕ : X E , intoa metric space ( E, d ), is bounded if each ϕ ( x ), x ∈ X , is a bounded subset of( E, d ). In these terms, the property in Proposition 2.15 that “Φ has a continuous ε -selection for some ε >
0” was used to construct a bounded-valued l.s.c. selection ϕ : X → F c ( E ) for Φ. Regarding this, let us mention the following property ofcountable paracompactness, see [83, Lemma 2.1] and [88, Lemma 1.2]. Proposition 2.16.
Let E be a normed space and X be a countably paracompactspace. Then every l.s.c. mapping Φ : X → F c ( E ) has a continuous selectionif and only if every bounded l.s.c. mapping ϕ : X → F c ( E ) has a continuousselection. This brings the following characterisation of countably paracompact PF-normalspaces, compare with Proposition 2.15.
Theorem 2.17.
A space X is countably paracompact and PF-normal if and onlyif for every normed space E , every l.s.c. mapping Φ : X → F c ( E ) with Φ( x ) beingeither compact or finite-dimensional, has continuous selection.Proof. In the one direction, this is Theorem 2.5 and Proposition 2.16 becauseeach closed bounded subset of a finite-dimensional normed space is compact, seethe proof of Proposition 2.15. Conversely, if X has the selection property in thetheorem, then it is PF-normal (by Theorem 2.5). By taking E to be the real line R and using [54, Theorem 3.1 ′′ ], X is also countably paracompact. (cid:3) A function ξ : X → R is lower ( upper ) semi-continuous if the set { x ∈ X : ξ ( x ) > r } (respectively, { x ∈ X : ξ ( x ) < r } )is open in X , for every r ∈ R . If ( E, d ) is a metric space, ϕ : X E and η : X → (0 , + ∞ ), then we shall say that g : X → E is an η -selection for ϕ if g ( x ) ∈ O η ( x ) ( ϕ ( x )), for every x ∈ X . According to Dowker-Katˇetov’s insertiontheorem [16, 50], see also [24, 5.5.20(a)], a space X is normal and countablyparacompact if and only if for every pair ξ, η : X → R of functions such that ξ is upper semi-continuous, η is lower semi-continuous and ξ < η , there existsa continuous function f : X → R with ξ < f < η . The following selectioninterpretation of this insertion property was shown in [39, Theorem 4.3]. Theorem 2.18 ([39]) . A space X is countably paracompact and collectionwisenormal if and only if for every Banach space E , l.s.c. mapping Φ : X → C ′ c ( E ) ,lower semi-continuous function η : X → (0 , + ∞ ) and continuous η -selection g : X → E for Φ , there exists a continuous selection f : X → E for Φ with k f ( x ) − g ( x ) k < η ( x ) , for all x ∈ X . Without the assumption of countable paracompactness, the following similarcharacterisation holds, see [39, Proposition 4.2].
Proposition 2.19 ([39]) . Let X be a collectionwise normal space, E be a Banachspace, Φ : X → C ′ c ( E ) be l.s.c., η : X → (0 , + ∞ ) be continuous and g : X → E be a continuous η -selection for Φ . Then Φ has a continuous selection f : X → E with k f ( x ) − g ( x ) k ≤ η ( x ) , for all x ∈ X . Motivated by this, the following question was posed in [39, Question 3].
Question 6 ([39]) . Let X be a collectionwise normal space, E be a Banach space,Φ : X → C ′ c ( E ) be an l.s.c. mapping and g : X → E be a continuous η -selectionfor Φ, for some lower semi-continuous function η : X → (0 , + ∞ ). Does Φ have acontinuous selection f : X → E with k f ( x ) − g ( x ) k ≤ η ( x ), for all x ∈ X ? Remark 2.20.
Let us point out that the answer to Question 6 is “Yes” if so isthe answer to Question 1. Indeed, let Φ, η and g be as in Question 6. Thenthe mapping O η [ g ] : X E , defined by O η [ g ]( x ) = O η ( x ) ( g ( x )), x ∈ X , has an ELECTIONS AND HIGHER SEPARATION AXIOMS 11 open graph [37, Proposition 2.1], see also Proposition 2.11. Hence, the mapping ϕ = Φ ∧ O η [ g ] remains convex-valued and l.s.c., by Propositions 2.9 and 2.10.Moreover, g ( x ) / ∈ ϕ ( x ) implies that Φ( x ) = E and, therefore, ϕ ( x ) is compactbeing a closed subset of the compact set Φ( x ). Evidently, if f : X → E is acontinuous selection for ϕ , then k f ( x ) − g ( x ) k ≤ η ( x ) for all x ∈ X . (cid:3) The property stated in Question 6 can be considered as a selection interpreta-tion of the classical Katˇetov-Tong insertion theorem [50, 51, 91, 92], see also [24,1.7.15(b)], that a space X is normal if and only if for every pair ξ, η : X → R of functions such that ξ is upper semi-continuous, η is lower semi-continuous and ξ ≤ η , there exists a continuous function f : X → R with ξ ≤ f ≤ η .
3. Selections and Continuity-Like Properties3.1. Selection Factorisation Properties.
Here, we briefly discuss two para-compact-like properties of set-valued mappings which offer a natural generalisationof Theorems 2.3 and 2.4. They are based on the following idea of factorising set-valued mappings. For a metrizable space Y and a mapping Φ : X Y , we say thata triple ( Z, h, ϕ ) is an l.s.c. weak-factorisation for
Φ [13, 69] if Z is a metrizablespace with weight w ( Z ) ≤ w ( Y ), h : X → Z is continuous and ϕ : Z Y is l.s.c.such that ϕ ◦ h : X Y is a set-valued selection for Φ.In what follows, using the convex hull operator A → conv( A ) of a linear space E , to each mapping Φ : X E we will associate the mapping conv[Φ] : X E ,defined by conv[Φ]( x ) = conv(Φ( x )), x ∈ X . It is well known that lower semi-continuity is preserved by passing to the mapping conv[Φ], [54, Proposition 2.6]. Proposition 3.1 ([54]) . Let E be a normed space and Φ : X E be an l.s.c.mapping. Then the mapping conv[Φ] : X E is also l.s.c. We now have the following general reduction of the selection problem for set-valued mappings.
Proposition 3.2.
Let X be a space, E be a Banach space and Φ : X → F c ( E ) be a mapping which admits an l.s.c. weak-factorisation. Then Φ has a continuousselection.Proof. Let (
Z, h, ϕ ) be an l.s.c. weak-factorisation for Φ. By Propositions 2.9and 3.1, the associated mapping conv[ ϕ ] : Z → F c ( E ) remains l.s.c. Hence, byTheorem 2.4, it has a continuous selection g : Z → E . Then g ◦ h : X → E is a continuous selection for Φ because conv[ ϕ ] ◦ h : X → F c ( E ) is a set-valuedselection for Φ. (cid:3) Considering l.s.c. weak-factorisations in the setting of Theorems 2.3 and 2.4,Nedev [69] (see also [13]) defined the following property. A mapping Φ : X Y is said to have the Selection Factorisation Property (called s.f.p. , for short) if for every closed subset F ⊂ X and every locally finite collection U of open subsets of Y such that Φ − [ U ] = { Φ − [ U ] : U ∈ U } covers F , there exists a locally finite andopen (in F ) cover of F which refines Φ − [ U ]. The importance of s.f.p. mappingsis evident from the following two observations. Example 3.3 ([69]) . Let Φ : X → F ( Y ) be an l.s.c. mapping, where Y is ametrizable space. Then Φ has the s.f.p. provided X is w ( Y )-paracompact, orwhen X is w ( Y )-collectionwise normal and Φ : X → C ′ ( Y ). Theorem 3.4 ([13, 69]) . Let X be a normal space and Y be a completely metriz-able space. Then each s.f.p. mapping Φ : X → F ( Y ) has an l.s.c. weak-factori-sation. To extend Theorem 3.4 to set-valued mappings defined on arbitrary spaces, thefollowing similar property was defined in [35]. Let (
Y, ρ ) be a metric space. Amapping Φ : X Y is said to be lower semi-factorisable relatively ρ , or ρ -l.s.f. , iffor every closed subset F ⊂ X , every ε > s : F → Y for Φ ↾ F , there exists a locally finite (in F ) covering U of F of cozero-sets of F , and a map κ : U → F such that | U | ≤ w ( Y ) and s ( κ ( U )) ∈ O ε (Φ( x )) , for every x ∈ U ∈ U .Here are two important properties of ρ -l.s.f. mappings. Example 3.5 ([35]) . Let X be a normal space, ( Y, ρ ) be a metric space andΦ : X Y be an s.f.p. mapping. Then Φ is ρ -l.s.f. Theorem 3.6 ([35]) . Let X be a space, ( Y, ρ ) be a complete metric space and Φ : X → F ( Y ) be a ρ -l.s.f. mapping. Then Φ has an l.s.c. weak-factorisation. Based on Theorem 3.6, the ρ -l.s.f.mappings deal with several other selection theorems which are similar to Theorems2.3 and 2.4, but any restriction on the domain is removed at the expense ofstrengthening the continuity of the set-valued mappings. This is discussed below.A mapping Φ : X Y is called continuous if it is both l.s.c. and u.s.c. Here,Φ is u.s.c. , or upper semi-continuous , if the set if Φ − [ F ] is closed in X for everyclosed F ⊂ Y ; equivalently, if Φ [ U ] = (cid:8) x ∈ X : Φ( x ) ⊂ U (cid:9) is open in X , forevery open U ⊂ Y .Let ( Y, ρ ) be a metric space. A mapping Φ : X Y is ρ -l.s.c. ( ρ -u.s.c. ) if forevery ε >
0, each point p ∈ X has a neighbourhood V such that Φ( p ) ⊂ O ε (Φ( x ))(respectively, Φ( x ) ⊂ O ε (Φ( p ))), for every x ∈ V . Metric semi-continuity offersother interpretations of continuity. Namely, a mapping Φ : X Y is called(i) ρ -continuous if it is both ρ -l.s.c. and ρ -u.s.c.;(ii) ρ -proximal continuous if it is both l.s.c. and ρ -u.s.c.; ELECTIONS AND HIGHER SEPARATION AXIOMS 13 (iii) proximal continuous if it is d -proximal continuous for some metric d on Y ,which is topologically equivalent to ρ .In the realm of set-valued mappings with a metrizable range, every continuousor ρ -continuous mapping is ρ -proximal continuous, but the converse is not true,see [35, Proposition 2.5]. Regarding continuity and ρ -continuity, it is well knownthat these properties coincide precisely when the range is a compact metric space,see e.g. [35, Proposition 2.6].The continuous and ρ -continuous mappings fit naturally into the selection the-ory. Selection theorems for ρ -continuous mappings with paracompact (or evenarbitrary) domain were obtained in [12, 56], while selection results for contin-uous mappings with arbitrary or (collectionwise) normal domain were obtainedin [11, 25]. Subsequently, these results were extended to proximal continuousmappings based on the following example, see [35, Example 4.3]. Example 3.7 ([35]) . If X is a space and ( Y, ρ ) is a metric space, then eachproximal continuous mapping Φ : X Y is ρ -l.s.f.A subset A ⊂ X is P λ -embedded in a space X , where λ is an infinite cardinalnumber, if for every locally finite cozero-set cover W of A of cardinality | W | ≤ λ ,there exists a locally finite cozero-set cover U of X such that W is refined by U ↾ A = { U ∩ A : U ∈ U } . The notion “ P λ -embedded” in this sense is the sameas “ P λ -embedded” in the sense of Shapiro [82], which was introduced by Arens[4] under the name “ λ -normally embedded” (see [82]). It is well known that everycontinuous map from a P λ -embedded subset A of a space X into a Banach space E of weight w ( E ) ≤ λ , is continuously extendable to the whole of X (Al´o andSennott [3], Morita [66], Przymusi´nski [75]). However, in the setting of arbitraryspaces, this extension property cannot be covered by the framework of continuousset-valued mapping. Indeed, for a Banach space E , a P w ( E ) -embedded set A ⊂ X and a continuous map g : A → E , the mapping Φ g : X → F ( E ) defined as inProposition 2.2 may fail to be even l.s.c.To rectify this, the following approach was offered in [35]. A map g : A → Y ,where A ⊂ X , is called A -regular [35] if for every locally finite cozero-set cover V of Y , there exists a locally finite cozero-set cover U of X such that g ( U ) refines V . A continuous g : A → Y is A -regular, whenever A is P w ( Y ) -embedded in X .Moreover, the restriction f ↾ A is A -regular, for every continuous map f : X → Y into a metrizable space Y . The following improved version of Example 3.7 wasobtained in [35, Example 4.4]. Example 3.8 ([35]) . Let X be a space, ( Y, ρ ) be a metric space, Φ : X → F ( Y )be proximal continuous, A ⊂ X and g : A → Y be an A -regular selection forΦ ↾ A . Define Φ g : X → F ( Y ) by Φ g ( x ) = { g ( x ) } if x ∈ A , and Φ g ( x ) = Φ( x )otherwise. Then the mapping Φ g is ρ -l.s.f. Combining this example with Proposition 3.2 and Theorem 3.6, the followinggeneral result was obtained in [35, Corollary 6.2].
Theorem 3.9 ([35]) . Let X be a space, E be a Banach space, Φ : X → F c ( E ) beproximal continuous, A ⊂ X and g : A → E be a continuous selection for Φ ↾ A .Then g can be extended to a continuous selection for Φ if and only if g can beextended to a continuous map f : X → E . The following result was obtained in[41, Lemma 4.2].
Theorem 3.10 ([41]) . Let X be a collectionwise normal space, E be a Banachspace, Φ : X → F ( E ) be proximal continuous and ϕ : X → F c ( E ) be an l.s.c.selection for Φ such that ϕ ( x ) is compact, whenever ϕ ( x ) = Φ( x ) . Then ϕ has acontinuous selection. Let us explicitly remark that the proof of Theorem 3.10 is based on the factthat, in this case, the mapping ϕ : X → F c ( E ) has the s.f.p. This idea is extendedin the following similar result. Theorem 3.11.
Let X be a collectionwise normal space, E be a Banach space, Φ : X → F c ( E ) be l.s.c. and A ⊂ X be a closed subset such that Φ ↾ X \ A isproximal continuous. Then each continuous selection g : A → E for Φ ↾ A can beextended to a continuous selection for Φ .Proof. Let g : A → E be a continuous selection for Φ ↾ A . Define Φ g : X → F c ( E )as in Example 3.8, namely Φ g ( x ) = { g ( x ) } if x ∈ A and Φ g ( x ) = Φ( x ) otherwise.The proof consists of showing that Φ g has the s.f.p. To this end, take a closed set F ⊂ X , and a locally finite family U of open subsets of E with F ⊂ S Φ − g [ U ].Since X is collectionwise normal, by Theorem 2.1, g can be extended to a con-tinuous map h : X → E . Then the family V = { Φ − g [ U ] ∩ h − ( U ) : U ∈ U } isopen and locally finite in X , and refines Φ − g [ U ]. Moreover, F = F ∩ A ⊂ S V because h ↾ A = g . Set F = F \ S V and ϕ = Φ g ↾ X \ A , so that F ⊂ S ϕ − [ U ].Since ϕ = Φ ↾ X \ A is proximal continuous, by [35, Theorem 3.1], ϕ − [ U ] isrefined by a σ -discrete (in X \ A ) family W of cozero-sets of X \ A such that S W = S ϕ − [ U ]. Since X is normal, there exists an open set G ⊂ X with F ⊂ G ⊂ G ⊂ S ϕ − [ U ] = S W . Finally, using [64, Theorem 1.2] and [65,Theorem 1.2], take an open and locally finite (in X \ A ) cover W of G , whichrefines W . Then the family V = { W ∩ G : W ∈ W } is open and locally finite in X . Moreover, it refines Φ − g [ U ] and covers F because F ⊂ G = [ V ⊂ [ W = [ ϕ − [ U ] ⊂ [ Φ − g [ U ] . Accordingly, V = V ∪ V is a locally finite family of open subset of X , whichrefines Φ − g [ U ] and covers F . Hence, Φ g has the s.f.p. and by Proposition 3.2 and ELECTIONS AND HIGHER SEPARATION AXIOMS 15
Theorem 3.4, it also has a continuous selection. Thus, g can be extended to acontinuous selection for Φ. (cid:3) Theorems 3.10 and 3.11 are naturally related to C ′ ( E )-valued mappings. Turn-ing to this, to each mapping Φ : X → F ( E ) we will associate the sets(3.1) C Φ = { x ∈ X : Φ( x ) ∈ C ( E ) } and D Φ = X \ C Φ . Here are some properties of these sets in the setting of C ′ ( E )-valued l.s.c. map-pings. Proposition 3.12.
Let E be a metrizable space and Φ : X → C ′ ( E ) be an l.s.c.mapping. Then (i) Φ is continuous at each point x ∈ D Φ ;(ii) C Φ is an F σ -set ;(iii) If Φ[ C Φ ] = S x ∈ C Φ Φ( x ) is not dense in E , then C Φ is closed in X .Proof. If p ∈ D Φ , then Φ [Φ( p )] = Φ [ E ] = X is open. This shows (i). If E iscompact, then D Φ = ∅ . If not, take a countable locally finite open cover V of E ,which has no finite subcover. Then D Φ = T Φ − [ V ], which shows (ii). To showfinally (iii), suppose that U ⊂ E is a nonempty open set with U ∩ Φ( x ) = ∅ , forevery x ∈ C Φ . Then D Φ = Φ − [ U ] is open, hence C Φ is closed. (cid:3) Based on this proposition and Theorems 3.10 and 3.11, see also Question 1,we have the following question relating the set C Φ to the selection problem forcollectionwise normal spaces. Question 7.
Let X be a collectionwise normal space, E be a Banach space andΦ : X → F c ( E ) be an l.s.c. mapping, which is continuous at each point p ∈ X with Φ( p ) / ∈ C ( E ). Does Φ have a continuous selection?Let X be a space, E be a Banach space, Φ : X → F c ( E ) be an l.s.c. mappingand D Φ be as in (3.1). If Φ ↾ D Φ is continuous, then it has a continuous selection g : D Φ → E [35, Theorem 6.1], see also Theorem 3.9. Since E is a Banach space(hence, a complete metric space), g can be extended to a continuous map onsome G δ -subset of X containing D Φ , see [24, Theorem 4.3.21]. Thus, accordingto Proposition 2.7, we get the following consequence. Corollary 3.13.
Let X be a space, E be a Banach space and Φ : X → F c ( E ) bean l.s.c. mapping such that Φ ↾ D Φ is continuous. Then Φ ↾ H has a continuousselection for some G δ -subset H ⊂ X with D Φ ⊂ H . On the other hand, collectionwise normality is hereditary with respect to F σ -sets [80, Theorem 1.3]. So, complementary to Corollary 3.13 is the followingconsequence of Theorems 2.3 (see also Theorem 2.5) for the set C Φ = X \ D Φ . Corollary 3.14.
Let X be a collectionwise normal space, E be a normed spaceand Φ : X → F c ( E ) be an l.s.c. mapping. Then Φ ↾ Z has a continuous selection,for every F σ -set Z ⊂ X with Z ⊂ C Φ . Corollaries 3.13 and 3.14 give a good illustration of Question 7 showing that onecan construct two partial continuous selections for Φ on complementary subsetsof the domain. Hence, the question is if one can use these partial selections, orother information, to construct a continuous selection for the mapping Φ itself.However, it is not so likely that any one of these partial selections can be extendedto a continuous selection for Φ.
Proposition 3.15.
Let
Φ : X → F ( E ) be an l.s.c. mapping, g : X → E be acontinuous map and A = { x ∈ X : g ( x ) ∈ Φ( x ) } . If g ↾ A can be extended to acontinuous selection f : X → E for Φ , then A must be closed.Proof. Follows from the fact that, in this case, A = { x ∈ X : g ( x ) = f ( x ) } . (cid:3) For a mapping Φ : X E and asubset A ⊂ X , let Φ[ A ] = S x ∈ A Φ( x ) (see Proposition 3.12). Proposition 3.16.
Let
Φ : X E be an l.s.c. mapping and A ⊂ X . Then theset Z = n x ∈ X : Φ( x ) ⊂ Φ[ A ] o is closed in X . In particular, Φ[ A ] is dense Φ[ X ] whenever A is dense in X .Proof. Simply observe that X \ Z = Φ − h E \ Φ[ A ] i is open. (cid:3) This brings the following refined version of Question 7.
Question 8.
Under the conditions of Question 7, suppose further that eachnonempty open set U ⊂ X contains a point p ∈ U with Φ( p ) ∈ C ( E ), i.e.that C Φ is dense in X , see (3.1). Does Φ have a continuous selection?Let us remark that if the answer to Question 8 is “Yes”, then by Theorem 3.11,so is the answer to Question 7. Finally, here is a bit more general question. Question 9.
Let X be a collectionwise normal space, E be a Banach space andΦ : X → F c ( E ) be an l.s.c. mapping such that each nonempty open set U ⊂ X contains a point p ∈ U with Φ( p ) ∈ C ( E ). Does Φ have a continuous selection? The following simple observation, obtained in [54,Example 1.3*], shows that the selection-extension problem for l.s.c. mappings isequivalent to the selection problem for these mappings.
Proposition 3.17 ([54]) . Let
Φ : X Y be l.s.c., A ⊂ X be closed and g : A → Y be a continuous selection for Φ ↾ A . Define Φ g : X Y by Φ g ( x ) = { g ( x ) } if x ∈ A , and Φ g ( x ) = Φ( x ) otherwise. Then the mapping Φ g also l.s.c. This implies the following interpretation of Theorem 2.4.
ELECTIONS AND HIGHER SEPARATION AXIOMS 17
Theorem 3.18.
Let X be paracompact, E be a Banach space, Φ : X → F c ( E ) be an l.s.c. mapping and Ω Φ be the collection of all continuous selections for Φ .Then Φ( x ) = { f ( x ) : f ∈ Ω Φ } , for every x ∈ X . Precisely the same interpretation holds for Theorem 2.3 as well.
Theorem 3.19.
Let X be a collectionwise normal space, E be a Banach space, Φ : X → C ′ c ( E ) be an l.s.c. mapping and Ω Φ be the collection of all continuousselections for Φ . Then Φ( x ) = { f ( x ) : f ∈ Ω Φ } , for every x ∈ X . This brings the following natural question; see Theorems 3.9 and 3.11, alsoProposition 3.12.
Question 10.
Let X be a collectionwise normal space, E be a Banach space andΦ : X → F c ( E ) be an l.s.c. mapping which has a continuous selection f : X → E .If A ⊂ X is closed and g : A → E is a continuous selection for Φ ↾ A , then is itpossible to extend g to a continuous selection for Φ?
4. Selections and Compact-Like Families4.1. A General Selection Problem.
The selection problem for collectionwisenormal spaces has two aspects. The one is simply the question for a particularset-valued mapping.
Question 11.
Let X be a collectionwise normal space, E be a Banach spaceand Φ : X → F c ( E ) be an l.s.c. mapping. When does there exist a continuousselection for Φ?The other question is about a particular family L ( E ) ⊂ F c ( E ) with the prop-erty that every l.s.c. mapping Φ : X → L ( E ) has a continuous selection. Question 12.
Let X be a collectionwise normal space and E be a Banach space.Find a large enough subfamily L ( E ) ⊂ F c ( E ) such that every l.s.c. mappingΦ : X → L ( E ) has a continuous selection?Question 12 is rather general, and to make sense natural restrictions are inplace. For instance, such a family should include Dowker’s extension theorem(Theorem 2.1) in the sense of the construction in Proposition 2.2. Therefore, onenatural requirement is that(4.1) C ′ c ( E ) ⊂ L ( E ) . Another natural condition may come from the construction of approximateselections in Proposition 2.19. Namely, this proposition can be rephrased in thefollowing way.
Proposition 4.1.
Let X be a collectionwise normal space, E be a Banach space, Φ : X → C ′ c ( E ) be l.s.c., η : X → (0 , + ∞ ) be continuous and g : X → E be acontinuous η -selection for ϕ . Then the mapping Φ ∧ O η [ g ] : X → F c ( E ) has acontinuous selection. Here, for a mapping ϕ : X E and a function η : X → (0 , + ∞ ), the mapping O η [ ϕ ] : X E is defined by O η [ ϕ ]( x ) = O η ( x ) ( ϕ ( x )), x ∈ X . If ϕ is l.s.c. (inparticular, a usual continuous map) and η is a lower semi-continuous function,then O η [ ϕ ] has an open graph [37, Proposition 2.1]; see Remark 2.20 where thiswas already used. Thus, by Propositions 2.9 and 2.10, the mapping Φ ∧ O η [ g ] inProposition 4.1 is also l.s.c. Hence, one can incorporate the property by consid-ering the following further condition on the collection L ( E ).(4.2) S ∩ O δ ( y ) ∈ L ( E ) , whenever S ∈ L ( E ) and y ∈ O δ ( S ) for some δ > Question 13.
Let X be a collectionwise normal space, E be a Banach space and L ( E ) ⊂ F c ( E ) be as in (4.1) and (4.2). Then is it true that each l.s.c. mappingΦ : X → L ( E ) has a continuous selection?Let us point out that Question 13 is related to Question 6, see Remark 2.20.Indeed, let Φ : X → C ′ c ( E ), η : X → (0 , + ∞ ) and g : X → E be as in Question6. Then Φ ∧ O η [ g ] : X → F c ( E ) is l.s.c. Moreover, if L ( E ) ⊂ F c ( E ) satisfies(4.1) and (4.2), then Φ ∧ O η [ g ] : X → L ( E ). Thus, if the answer to Question 13is “Yes”, then Φ ∧ O η [ g ] has a continuous selection, and the answer to Question6 will be “Yes” as well. Related to Question 12, the following interestingresult was obtained by Nedev and Valov [71].
Theorem 4.2 ([71]) . Let X be a normal space which is not countably paracompact, E be a Banach space, and L ( E ) ⊂ F c ( E ) be such that every l.s.c. mapping Φ : X → L ( E ) has a continuous selection. Then any decreasing sequence ofelements of L ( E ) has a nonempty intersection.Proof. We present the proof in [71, (b) of Theorem 1]. Contrary to the claim,suppose that L ( E ) has a strictly decreasing sequence { F n } with an empty in-tersection. So, for every n ∈ N , there is a point z n ∈ F n \ F n +1 . Then the set H = { z n : n ∈ N } is closed and discrete in F because T ∞ n =1 F n = ∅ . Next,for every y ∈ F , set m ( y ) = max { n ∈ N : y ∈ F n } and define an l.s.c. mapping ϕ : F → F ( H ) by ϕ ( y ) = (cid:8) z k : k ≥ m ( y ) (cid:9) , y ∈ F . Since F is paracompact (be-ing metrizable), by a result of Michael [57], ϕ has a u.s.c. selection ψ : F → C ( H ).On the other hand, X is normal but not countably paracompact. Hence, it hasan increasing open cover { U n } which doesn’t admit a closed cover { P n } with ELECTIONS AND HIGHER SEPARATION AXIOMS 19 P n ⊂ U n , n ∈ N . Whenever x ∈ X , set n ( x ) = min { n ∈ N : x ∈ U n } , and definean l.s.c. mapping Φ : X → L ( E ) by Φ( x ) = F n ( x ) , x ∈ X . By hypothesis, Φhas a continuous selection f : X → E . Finally, consider the composite mapping θ = ψ ◦ f : X → C ( H ), which is clearly u.s.c. So, for every n ∈ N , the set P n = θ − [ { z , . . . , z n } ] = f − ( ψ − [ { z , . . . , z n } ]) is closed in X , and P n ⊂ U n be-cause { z , . . . z n } ⊂ F n \ F n +1 . Indeed, x ∈ P n implies that ψ ( f ( x )) ⊂ { z , . . . , z n } and, therefore, m ( f ( x )) ≤ n , by the definition of ϕ . Accordingly, f ( x ) / ∈ F n +1 and, by the definition of Φ, we get that n ( x ) ≤ n . Thus, x ∈ U n ( x ) ⊂ U n and P n ⊂ U n . Since { P n } is covering X , this is impossible. (cid:3) The following question is a partial case of a question stated in [71].
Question 14 ([71]) . Let X be a collectionwise normal space, E be a Banachspace, and L ( E ) ⊂ F c ( E ) be such that C ′ c ( E ) ⊂ L ( E ) and any decreasingsequence of elements of L ( E ) has a nonempty intersection. Then, is it true thateach l.s.c. mapping Φ : X → L ( E ) has a continuous selection? Remark 4.3.
An elegant alternative proof of Theorem 4.2 was offered by thereferee. Namely, take a decreasing sequence { F n } of elements L ( E ). Since X isnormal but not countably paracompact, there exists a decreasing sequence { P n } of closed subsets of X such that P = X , T ∞ n =1 P n = ∅ and T ∞ n =1 U n = ∅ , forevery sequence { U n } of open subsets of X with P n ⊂ U n , n ∈ N , see [24, Corollary5.2.2]. Next, for every x ∈ X , let n ( x ) = max { n ∈ N : x ∈ P n } which is awell-defined element of N because T ∞ n =1 P n = ∅ . Finally, define Φ : X → L ( E )by Φ( x ) = F n ( x ) , x ∈ X . Then Φ is l.s.c. and by hypothesis, it has a continuousselection f : X → E . Since T ∞ n =1 F n is closed and P n ⊂ f − (cid:0) O /n ( F n ) (cid:1) , for every n ∈ N , we get that f − ∞ \ n =1 F n ! = f − ∞ \ n =1 O /n ( F n ) ! = ∞ \ n =1 f − (cid:0) O /n ( F n ) (cid:1) = ∅ . Accordingly, T ∞ n =1 F n = ∅ as required. (cid:3) The property in Theorem 4.2has the following natural interpretation in the setting of Banach spaces.
Theorem 4.4 ([98]) . A normed space E is a reflexive Banach space if and onlyif every decreasing sequence of nonempty closed bounded convex subsets of E hasa nonempty intersection. For reflexive Banach spaces, the following interesting result was obtained byStoyan Nedev [70].
Theorem 4.5 ([70]) . Whenever E is a reflexive Banach space, each l.s.c. mapping Φ : ω → F c ( E ) has a continuous selection. Here, ω is the first uncountable ordinal endowed with the order topology. Thisresult was further generalised in [14] by replacing ω with an arbitrary suborder-able space. Theorem 4.6 ([14]) . If X is a suborderable space and E is a reflexive Banachspace, then each l.s.c. mapping Φ : X → F c ( E ) has a continuous selection.Suborderable spaces are precisely the subspaces of orderable spaces, and are alsocalled generalised ordered . Every suborderable space is countably paracompactand collectionwise normal, but not necessarily paracompact. For instance, ω isnot paracompact. Based on this, the following general question was stated in [14];it is known as Choban-Gutev-Nedev conjecture. Question 15 ([14]) . Let X be a countably paracompact and collectionwise normalspace, E be a Hilbert (or reflexive Banach) space and Φ : X → F c ( E ) be an l.s.c.mapping. Does Φ have a continuous selection? Let E be a normed linear space. A selection f : X → E for a mapping Φ : X E is called minimal with respect to the norm k · k of E , or norm-minimal , see [5], if(4.3) k f ( x ) k = min (cid:8) k y k : y ∈ Φ( x ) (cid:9) , for every x ∈ X. A norm k . k on E is called locally uniformly rotund , abbreviated LUR , if for each y ∈ E and sequence { y n } ⊂ E ,(4.4) ( lim n →∞ k y n k = k y k , andlim n →∞ k y n + y k = 2 k y k , implies lim n →∞ k y n − y k = 0 . If E is a normed space equipped with an LUR norm, then every nonempty closedconvex subset of E has a unique point with a minimal norm, see [14, Lemma 4.1].Accordingly, we have the following observation. Proposition 4.7.
Let E be a normed space equipped with an LUR norm. Theneach mapping Φ : X → F c ( E ) has a unique norm-minimal selection. Regarding continuity of norm-minimal selections, the following characterisationwas obtained in [40, Theorem 4.1]. In this theorem, B is the closed unit ball of anormed space E equipped with a norm k · k . Theorem 4.8 ([40]) . Let X be a space and E be a normed space equipped withan LUR norm. Then for an l.s.c. mapping Φ : X → F c ( E ) , the following twoconditions are equivalent :(a) Φ − [ ε B ] is closed in X , for every ε > . (b) Φ admits a continuous norm-minimal selection. As for normed spaces which admit an equivalent LUR norm, let us explicitlystate the famous Troyanski’s renorming theorem [95, Theorem 1].
ELECTIONS AND HIGHER SEPARATION AXIOMS 21
Theorem 4.9 ([95]) . Every reflexive Banach space admits a topologically equiva-lent LUR norm.
Based on this theorem and norm-minimal selections, the following interestingresult about Question 15 was obtained by Shishkov [86, Proposition 1.1].
Theorem 4.10 ([86]) . Let E be a reflexive Banach space and X be a space suchthat each weak θ -cover of X has an open locally finite refinement. Then everyl.s.c. mapping Φ : X → F c ( E ) has a continuous selection. Here, a cover U of X is called a weak θ -cover if U is a countable union of openfamilies U k , k ∈ N , such that for each x ∈ X , there exists some k ( x ) ∈ N forwhich the family U k ( x ) has a positive finite order at x , namely0 < (cid:12)(cid:12)(cid:8) U ∈ U k ( x ) : x ∈ U (cid:9)(cid:12)(cid:12) < + ∞ . The following example was given in the same paper of Shishkov, see [86, Theo-rem 1.2].
Example 4.11 ([86]) . If X is a countably paracompact and hereditarily collec-tionwise normal space, then each weak θ -cover of X has an open locally finiterefinement.Example 4.11 implies that Theorem 4.10 is a natural generalisation of Theorem4.6 because each suborderable space is countably paracompact and hereditarilycollectionwise normal. In fact, Theorem 4.10 is a potential candidate for theaffirmative solution of Question 15 in view of the following characterisation ofcountably paracompact collectionwise normal spaces claimed in [90]. Theorem 4.12 ([90]) . A space X is countably paracompact and collectionwisenormal if and only if each weak θ -cover of X has an open locally finite refinement. However, as pointed out in [86], the proof of Theorem 4.12 in [90] is incomplete,which suggests the following separate question.
Question 16.
Let X be a countably paracompact collectionwise normal spaceand U be a weak θ -cover of X . Is it true that U has an open locally finiterefinement? In case of Hilbert spaces, Question 15was resolved in the affirmative by Ivailo Shishkov in 2005, his paper with the finalsolution appeared in print in [88].
Theorem 4.13 ([88]) . A space X is countably paracompact and collectionwisenormal iff for every Hilbert space E , every l.s.c. mapping Φ : X → F c ( E ) has acontinuous selection. The role of countable paracompactness in Theorem 4.13 is the equivalencestated in Proposition 2.16, while the essential selection property of collectionwisenormality was obtained in [88, Theorem 1.3].
Theorem 4.14 ([88]) . If X is a collectionwise normal space and E is a Hilbertspace, then each bounded l.s.c. mapping Φ : X → F c ( E ) has a continuous selec-tion. Here are two consequences of Theorem 4.14, which may shed some light on therole of Hilbert spaces in the selection problem for collectionwise normal spaces.The first one shows that the answer to Question 5 is “Yes” provided the range E is a Hilbert space. Corollary 4.15.
Let X be a collectionwise normal space, E be a Hilbert spaceand Φ : X → F c ( E ) be an l.s.c. mapping. If Φ has a continuous ε -selection forsome ε > , then Φ also has a continuous selection.Proof. Let g : X → E be a continuous ε -selection for Φ, for some ε > ∧ O ε [ g ] : X → F c ( E )which is bounded-valued. Hence, by Theorem 4.14, it has a continuous selection f : X → E . Evidently, f is also a selection for Φ. (cid:3) The other consequence should be compared with the characterisation of PF-normality in Theorem 2.5.
Corollary 4.16.
Let X be a space such that for every Hilbert space E , everybounded l.s.c. mapping Φ : X → F c ( E ) has a continuous selection. Then X iscollectionwise normal.Proof. Let D be a discrete collection of nonempty closed subsets of X , and ℓ ( D )be the Hilbert space of all functions y : D → R with P D ∈ D [ y ( D )] < ∞ , wherethe linear operations are defined pointwise and the norm is k y k = pP D ∈ D [ y ( D )] .Set A = S D , and define a map g : A → ℓ ( D ) by g ( x ) = χ D : D → { , } ⊂ R to be the characteristic function of the unique D ∈ D with x ∈ D . Since D isclosed and discrete, A is a closed subset of X and g is continuous. Moreover, g takes values in the closed unit ball B of ℓ ( D ) because k χ D k = 1, for each D ∈ D . Finally, let Φ g : X → F c ( B ) ⊂ F c ( ℓ ( D )) be as in Proposition 2.2with Y replaced by B . Then Φ g is both l.s.c. and is bounded-valued. Hence, byhypothesis, it has a continuous selection f : X → ℓ ( D ), which is a continuousextension of g , by Proposition 2.2. Since { χ D : D ∈ D } is a closed discrete setin ℓ ( D ) (actually, uniformly discrete with respect to the norm), there exists adiscrete collection { U D : D ∈ D } of open subsets of ℓ ( D ) with χ D ∈ U D , for each D ∈ D . Then U D = f − ( U D ), D ∈ D , is a discrete collection of open subsets of X such that D ⊂ U D , for each D ∈ D . Accordingly, X is collectionwise normal. (cid:3) ELECTIONS AND HIGHER SEPARATION AXIOMS 23 ℓ p ( A ) -spaces. For a set A and p ≥
1, let ℓ p ( A ) be theBanach space of all functions y : A → R with P α ∈ A | p ( α ) | p < ∞ , where thelinear operations are defined pointwise and the norm is k y k p = (cid:0)P α ∈ A | y ( α ) | p (cid:1) p .Since every Hilbert space is isomorphic to ℓ ( A ) for some set A , Theorem 4.13can be restated in the following terms. Theorem 4.17 ([88]) . For a space X , the following are equivalent :(a) X is countably paracompact and collectionwise normal. (b) For every set A , every l.s.c. mapping Φ : X → F c ( ℓ ( A )) has a continu-ous selection. Similarly, the characterisation of paracompactness in Theorem 2.4 can be re-stated in terms of the Banach space ℓ ( A ). The following theorem is actuallyreassembling the proof of Theorem 2.4. Theorem 4.18 ([54]) . For a space X , the following are equivalent :(a) Every open cover U of X has a locally finite open refinement. (b) For every set U , every l.s.c. mapping Φ : X → F c ( ℓ ( U )) has a contin-uous selection. (c) Every open cover U of X has an index-subordinated partition of unity. Here, a collection ξ U : X → [0 , U ∈ U , of continuous functions on a space X is a partition of unity if P U ∈ U ξ U ( x ) = 1, for each x ∈ X . A partition of unity { ξ U : U ∈ U } is index-subordinated to a cover U of X if X \ U ⊂ ξ − U (0), foreach U ∈ U . The following natural result may explain the relationship betweenpartitions of unity, index-subordinated to open covers U of X , and continuousselections f : X → ℓ ( U ), see Theorem 4.18. The result itself was obtained inthe implication (f) ⇒ (a) of [64, Theorem 1.2] and was explicitly stated in [23,Proposition 5.4]; the case of locally finite partitions of unity was obtained in theproof of [76, Theorem 1]. Lemma 4.19.
Let X be a space and ξ α : X → [0 , , α ∈ A , be a collection offunctions. Then { ξ α : α ∈ A } is a partition of unity on X if and only if thediagonal map ξ = ∆ α ∈ A ξ α : X → ℓ ( A ) is continuous and satisfies k ξ ( x ) k = 1 ,for every x ∈ X . Theorems 4.17 and 4.18 reveal an interesting role of the spaces ℓ ( A ) and ℓ ( A ) in the selection theory. In this regard, let us recall that all Banach spaces ℓ p ( A ), 1 ≤ p < + ∞ , are homeomorphic, and it is well known that ℓ ( A ) isnot reflexive, but each ℓ p ( A ), 1 < p < + ∞ , is reflexive. In fact, each infinite-dimensional reflexive Banach space is homeomorphic to ℓ ( A ), for some A , seee.g. [6, Theorem 4.1 in § Question 17.
Let X be a countably paracompact collectionwise normal space,and Φ : X → F c ( ℓ p ( A )) be an l.s.c. mapping for some A and 1 < p < + ∞ with p = 2. Does Φ have a continuous selection?Going back to Question 15, the interested reader may consult some of the papersof Shishkov ([83, 84, 85, 86, 87]), which contain several interesting ideas.
5. Selections and Finite-Dimensional Spaces5.1. Selection Extension and Approximation Properties.
Let n ≥ −
1. Afamily S of subsets of a space Y is equi- LC n [55] if every neighbourhood U of apoint y ∈ S S contains a neighbourhood V of y such that for every S ∈ S , everycontinuous map g : S k → V ∩ S of the k -sphere S k , k ≤ n , can be extended to acontinuous map h : B k +1 → U ∩ S of the ( k + 1)-ball B k +1 . A space S is C n if forevery k ≤ n , every continuous map g : S k → S can be extended to a continuousmap h : B k +1 → S . In these terms, a family S of subsets of Y is equi- LC − if itconsists of nonempty subsets; similarly, each nonempty subset S ⊂ Y is C − .A mapping Φ : X Y has the Selection Extension Property (or
SEP ) at aclosed subset A ⊂ X [59] if every continuous selection g : A → Y for Φ ↾ A canbe extended to a continuous selection for Φ. If g only extends to a continuousselection for Φ ↾ U for some neighbourhood U of A in X , then Φ is said to have the Selection Neighbourhood Extension Property (or
SNEP ) at A [59]. If this holdsfor any closed set of X , then we simply say that Φ has the SEP, or the SNEP.For a subset Z ⊂ X , we write dim X ( Z ) ≤ m to express that the coveringdimension dim( S ) ≤ m , for every S ⊂ Z which is closed in X , see [55]. Let usremark that for a normal space X , dim X ( Z ) ≤ m is valid if either dim( Z ) ≤ m or dim( X ) ≤ m . The following theorem was obtained in [55, Theorem 1.2] and iscommonly called the finite-dimensional selection theorem . Theorem 5.1 ([55]) . Let X be a paracompact space, A ⊂ X be a closed set with dim X ( X \ A ) ≤ n + 1 , Y be a completely metrizable space and S ⊂ F ( Y ) be anequi- LC n family. Then every l.s.c. mapping Φ : X → S has the SNEP at A . If,moreover, each S ∈ S is C n , then Φ also has the SEP at A . The proof of Theorem 5.1 in [55] is based on a uniform version of the sametheorem. To this end, let us recall that a family S of subsets of a metric space( Y, ρ ) is uniformly equi- LC n , where n ≥ −
1, if for every ε > δ ( ε ) > S ∈ S , every continuous map of the k -sphere ( k ≤ n ) in S of diameter < δ ( ε ) can be extended to continuous map of the ( k + 1)-ball intoa subset of S of diameter < ε [55]. The relation with equi- LC n families is givenby the following embedding property stated in [55, Theorem 3.1], see also [55,Proposition 2.1] and [22, Theorem 1]. ELECTIONS AND HIGHER SEPARATION AXIOMS 25
Theorem 5.2 ([55]) . Let S ⊂ F ( Y ) be an equi- LC n family of subsets of acompletely metrizable space Y . Then S S can be embedded into a Banach space E so that S ⊂ F ( E ) is a uniformly equi- LC n family of subsets of E . The other reduction in the proof of Theorem 5.1 is that the properties “SNEP”and “SEP” are obtained by the following uniform selection approximation prop-erty [55, Theorem 4.1].
Theorem 5.3 ([55]) . Let ( Y, ρ ) be a complete metric space and S ⊂ F ( Y ) beuniformly equi- LC n . Then to every ε > there corresponds γ ( ε ) > with thefollowing property : If Φ : X → S is an l.s.c. mapping from a paracompact space X with dim( X ) ≤ n +1 , then for every continuous γ ( ε ) -selection g : X → Y for Φ ,there exists a continuous selection f : X → Y for Φ such that ρ ( f ( x ) , g ( x )) < ε ,for all x ∈ X . Moreover, if each S ∈ S is C n , then one can take γ (+ ∞ ) = + ∞ . As a common generalisation of Theorems2.4 and 5.1, the following two theorems were obtained in [59].
Theorem 5.4 ([59]) . Let X be a paracompact space, E be a Banach space, Z ⊂ X with dim X ( Z ) ≤ n + 1 , and Φ : X → F ( E ) be an l.s.c. mapping with Φ( x ) convexfor all x ∈ X \ Z , and with { Φ( x ) : x ∈ Z } uniformly equi- LC n . Then Φ has theSNEP. If, moreover, Φ( x ) is C n for every x ∈ Z , then Φ has the SEP. To state the other theorem, let us recall that a family S of subsets of a space Y is equi- LC n in Y [59] if every neighbourhood U of a point y ∈ Y contains aneighbourhood V of y such that for every S ∈ S , every continuous g : S k → V ∩ S ,for k ≤ n , can be extended to a continuous h : B k +1 → U ∩ S . Each family ofsubsets of Y which is equi- LC n in Y is also equi- LC n , but the converse is notnecessarily true. Here is a simple example. Example 5.5.
For every positive real number t >
0, let S t = { x ∈ R : | x | ≥ t } = ( −∞ , − t ] ∪ [ t, + ∞ ) . Then S = { S t : t > } is equi- LC n for all n ≥ −
1, but is not equi- LC in R . (cid:3) Finally, let us also recall that a metrizable space Y is an AR (respectively, ANR ) if it is a retract (respectively, neighbourhood retract) of every metric space E containing it as a closed subset. Theorem 5.6 ([59]) . Let X be a paracompact space, Y be a completely metrizableANR, Z ⊂ X with dim X ( Z ) ≤ n + 1 , and Φ : X → F ( Y ) be an l.s.c. mappingwith Φ( x ) = Y for all x ∈ X \ Z , and with { Φ( x ) : x ∈ Z } equi- LC n in Y . Then Φ has the SNEP. If, moreover, Y is an AR and Φ( x ) is C n for every x ∈ Z , then Φ has the SEP. We proceed with an example showing that Theorem 5.4 fails if in this theoremthe collection { Φ( x ) : x ∈ Z } is assumed to be only equi- LC n . Example 5.7.
Let D ⊂ R be the closed unit disk in R , and S be the unit circle.Also, let S y = { ( s, t ) : s + t = 1 and t ≥ y } , for every − < y ≤
1. DefineΦ : D → F ( R ) by letting for ( x, y ) ∈ D thatΦ( x, y ) = ((cid:8) ( x, y ) (cid:9) if ( x, y ) ∈ S , S y if ( x, y ) / ∈ S .Then Φ is l.s.c., but has no continuous selection because each such selection willbe a retraction r : D → S . However, each Φ( x, y ), ( x, y ) ∈ S , is convex beinga singleton. Moreover, the collection S = { S y : − < y ≤ } of arcs is equi- LC n for every n ≥ −
1, and each element of S is C n . Finally, we also have thatdim D ( D \ S ) = dim( D \ S ) = 2. (cid:3) The case when the family in Theorem 5.4 is assumed to be equi- LC n in E isnot covered by this example, which brings the following question. Question 18.
Is Theorem 5.4 still valid if in this theorem “uniformly equi- LC n ”is replaced by “equi- LC n in E ”?This also brings a similar question about Theorem 5.6 of whether this theoremis still valid if “equi- LC n in Y ” is replaced by “equi- LC n ”. This doesn’t seemlikely, but is not covered by Example 5.7.Let us remark that in the special case of n = −
1, Theorem 5.6 is covered byTheorem 5.4, see the remark after the proof of Theorem 5.6 in [59]. Moreover, if Y and Φ are as in Theorem 5.6, then by [59, Lemma 6.1], Y can be embedded asa closed subset in a Banach space E such that(i) Y is a uniform ANR (respectively, uniform AR ) of E , and(ii) { Φ( x ) : x ∈ Z } is uniformly equi- LC n in E .Here, a closed subset Y ⊂ E is a uniform ANR of E [58] if to every ε > δ ( ε ) > r : O δ ( ∞ ) ( Y ) → Y such that k z − r ( z ) k < ε ,whenever z ∈ O δ ( ε ) ( Y ). If one can take δ ( ∞ ) = ∞ (so that the domain of r isalways E ), then Y is called a uniform AR of E . Accordingly, Theorem 5.6 can bereformulated in the following way. Theorem 5.8.
Let X be a paracompact space, E be a Banach space, Y ⊂ E be aclosed subset of E which is a uniform ANR of E , Z ⊂ X with dim X ( Z ) ≤ n + 1 ,and Φ : X → F ( Y ) be an l.s.c. mapping with Φ( x ) = Y for all x ∈ X \ Z , andwith { Φ( x ) : x ∈ Z } uniformly equi- LC n . Then Φ has the SNEP. If, moreover, Y is a uniform AR of E and Φ( x ) is C n for every x ∈ Z , then Φ has the SEP. This makes Theorems 5.4 and 5.6 further similar in the following sense. If Φis as in Theorem 5.4, then the l.s.c. mapping ˜Φ = conv[Φ] : X → F c ( E ) has theSEP, see Theorem 3.18. If Φ is as in Theorem 5.8, then the constant mapping˜Φ( x ) = Y , x ∈ X , has the SNEP, and also the SEP provided Y is a uniform ELECTIONS AND HIGHER SEPARATION AXIOMS 27
AR of E . Moreover, in both cases, the pair (Φ , ˜Φ) of mappings has the followingproperties:Φ is an l.s.c. set-valued selection for ˜Φ;(5.1) dim X (cid:0) { x ∈ X : Φ( x ) = ˜Φ( x ) } (cid:1) ≤ n + 1;(5.2) (cid:8) Φ( x ) : x ∈ X and Φ( x ) = ˜Φ( x ) (cid:9) is uniformly equi- LC n in E .(5.3)This motivates the following further question. Question 19.
Let X be a paracompact space and E be a Banach space. Supposethat (Φ , ˜Φ) : X → F ( E ) is a pair of mappings as in (5.1), (5.2) and (5.3). DoesΦ have the SNEP provided so does ˜Φ? Similarly, does Φ have the SEP provided˜Φ has the SEP and Φ( x ) is C n , for every x ∈ X with Φ( x ) = ˜Φ( x )? The following selection in-terpretation of the Borsuk homotopy extension theorem [8] was obtained byMichael, see [56, Theorem 3.4].
Theorem 5.9 ([56]) . Let X be a paracompact space, A ⊂ X be a closed set with dim X ( X \ A ) ≤ n , ( Y, ρ ) be a complete metric space, and Φ : X × [0 , → F ( Y ) be a quasi-continuous mapping such that { Φ( p ) : p ∈ X × [0 , } is uniformlyequi- LC n . Then Φ has the SEP at X × { } ∪ A × [0 , . Here, a mapping Φ : X × [0 , Y , into a metric space ( Y, ρ ), is quasi-continuous if it is l.s.c. and for every ε >
0, each point of X × [0 ,
1] has aneighbourhood U such that Φ( x, s ) ⊂ O ε (Φ( x, t )), whenever ( x, s ) , ( x, t ) ∈ U with s ≤ t . The interested reader is referred to [33, 34, 36], where Theorem 5.9was refined and generalised in various directions.Let Φ : X × [0 , Y . A mapping H : X × [0 , → Y is a Φ -homotopy if it isa continuous selection for Φ, and H is a Φ -homotopy of f : X × { } → Y if it isalso a continuous extension of f . The mapping Φ : X × [0 , Y is said to havethe Selection Homotopy Extension Property at a subset A ⊂ X , or the SHEP at A , if whenever f : X × { } → Y is a continuous selection for Φ ↾ X × { } , everyΦ ↾ A × [0 , G : A × [0 , → Y of f ↾ A × { } can be extended to aΦ-homotopy H : X × [0 , → Y of f . For instance, the mapping Φ in Theorem5.9 has the SHEP at A .Theorem 5.9 has a nice interpretation for set-valued mappings Φ : X Y de-fined only on X . In this case, we will say that a mapping H : X × [0 , → Y isa Φ -homotopy if H ( x, t ) ∈ Φ( x ), for every x ∈ X and t ∈ [0 , f : X → Y will be called Φ -homotopic to a selection g : X → Y if f and g are homotopic by a Φ-homotopy H : X × [0 , → Y . In these terms, we havethe following consequence of Theorem 5.9. Corollary 5.10.
Let X be a paracompact space, A ⊂ X be a closed set with dim X ( X \ A ) ≤ n , Y be a completely metrizable space and Φ : X → F ( Y ) bean l.s.c. mapping such that { Φ( x ) : x ∈ X \ A } is equi- LC n in Y . Also, let g, h : A → Y be continuous selections for Φ ↾ A which are Φ ↾ A -homotopic. If oneof these selections can be extended to a continuous selection for Φ , then so doesthe other in such a way that both selection remain Φ -homotopic.Proof. Suppose that g can be extended to a continuous selection f : X → Y for Φ,and take a Φ ↾ A -homotopy G : A × [0 , → Y between g and h , say G ( x,
0) = g ( x )and G ( x,
1) = h ( x ), for every x ∈ A . For convenience, define a continuous map u : X × { } ∪ A × [0 , → E by u ( x, t ) = G ( x, t ) for ( x, t ) ∈ A × [0 , u ( x,
0) = f ( x ), x ∈ X . Next, define a mapping Φ u : X × [0 , → F ( Y ) byΦ u ( x, t ) = { u ( x, t ) } if ( x, t ) ∈ X × { } ∪ A × [0 , u ( x, t ) = Φ( x ) otherwise.Then the family { Φ u ( x, t ) : ( x, t ) ∈ X × [0 , } remains equi- LC n in Y . Hence, Y admits a complete compatible metric ρ so that this family is uniformly equi- LC n with respect to ρ , see Theorem 5.2. Moreover, Φ u is l.s.c. because so is Φ, seeProposition 3.17. In fact, it is easy to see that Φ u is quasi-continuous. Thus, byTheorem 5.9, u can be extended to a continuous selection H : X × [0 , → Y for Φ u . This H is a Φ-homotopy between f and a continuous extension of theselection h . (cid:3) A closed subset A ⊂ X of a space X is called a weak deformation retract of X if there exists a continuous r : X × [0 , → X such that r ( x,
1) = x for every x ∈ X , r ( x, ∈ A for every x ∈ X , r ( x,
0) = x for every x ∈ A .The following theorem was proved by Michael [56, Theorem 6.1]. Theorem 5.11 ([56]) . Let X be a paracompact space with dim( X ) ≤ n + 1 , ( Y, ρ ) be a complete metric space, S ⊂ F ( Y ) be a uniformly equi- LC n family, Φ : X → S be a ρ -continuous mapping and A ⊂ X be a weak deformation retractof X . Then every continuous selection g : A → Y for Φ ↾ A can be extended to acontinuous selection for Φ . Let us remark that, in contrast to Theorem 5.1, here there is no requirementthat each Φ( x ) is C n . Regarding the condition dim( X ) ≤ n + 1, the followingquestion was stated by Michael in [56]. Question 20 ([56]) . Does Theorem 5.11 remain true if dim( X ) ≤ n +1 is replacedby the weaker requirement that dim X ( X \ A ) ≤ n + 1? ELECTIONS AND HIGHER SEPARATION AXIOMS 29
As commented by Michael, see [56, Theorem 6.2], the answer to Question 20 is“Yes” provided the condition on A is strengthened to the existence of a continuous r : X × [0 , → Y such that r ( x,
1) = x for every x ∈ X , r ( x, ∈ A for every x ∈ X , r ( x, t ) ∈ A for every x ∈ A and 0 ≤ t ≤
6. Selections and Infinite-Dimensional Spaces6.1. Selections and Countable Dimensionality.
Another hybrid selectiontheorem representing a common generalisation of Theorems 2.4 and 5.1 was ob-tained in [59, Theorem 1.4].
Theorem 6.1 ([59]) . Let X be a paracompact space, A ⊂ X be a closed set with dim X ( X \ A ) ≤ n + 1 , Z ⊂ X \ A with dim X ( Z ) ≤ m + 1 , where m ≤ n , Y be acompletely metrizable space, and Φ : X → F ( Y ) be an l.s.c. mapping such that { Φ( x ) : x ∈ X \ Z } is equi- LC n in Y and { Φ( x ) : x ∈ Z } is equi- LC m in Y . Then Φ has the SNEP at A . If, moreover, Φ( x ) is C n for all x ∈ X \ Z and C m for all x ∈ Z , then Φ has the SEP at A . Subsequently, Theorem 6.1 was generalised by replacing Z with finitely manysuch sets, see [2]. The case of infinitely many sets seems to offer an interestingquestion, which is discussed below.The local dimension locdim( X ) of a space X was introduced by Dowker [18] asthe least number n such that each point of X is contained in an open set U withdim (cid:0) U (cid:1) ≤ n . It was shown in [18] that locdim( X ) ≤ dim( X ) for every normalspace X , but there exists a normal space X with locdim( X ) < dim( X ). Theorem 6.2 ([18]) . If X is a paracompact space, then locdim( X ) = dim( X ) . Subsequently, Wenner [99] generalised the local dimension and introduced theso called locally finite-dimensional spaces. A space X is locally finite-dimensional [99] if each point p ∈ X has a finite-dimensional neighbourhood. In these terms,a normal space X is locally finite-dimensional if each point p ∈ X is containedin an open set U ⊂ X with dim (cid:0) U (cid:1) < ∞ ; equivalently, if each p ∈ X has aneighbourhood U ⊂ X with dim X ( U ) < ∞ .A space X is countable-dimensional if it is a countable union of finite-dimensio-nal subsets [67]. A space X is strongly countable-dimensional if it is a count-able union of closed finite-dimensional subsets [67]. Each strongly countable-dimensional space is countable-dimensional, but the converse is not necessarilytrue [67, Example 5.2]. Each locally finite-dimensional metrizable space is stronglycountable-dimensional [99, Theorem 1]. Essentially the same proof remains validfor locally finite-dimensional paracompact spaces. Theorem 6.3 ([99]) . Every locally finite-dimensional paracompact space is stron-gly countable-dimensional.
On the other hand, let us remark that there exists a strongly countable-dimen-sional metrizable space which is not locally finite-dimensional [99, Theorem 4].Regarding selections and locally finite-dimensional spaces, the following natural“infinite-dimensional” version of Theorem 5.1 can be obtained following the proofof [54, Theorem 8.2], see also [60, Theorem 6.2].
Theorem 6.4.
Let X be a locally finite-dimensional paracompact space, Y bea completely metrizable space, and Φ : X → F ( Y ) be an l.s.c. mapping suchthat { Φ( x ) : x ∈ X } is equi- LC n for each n ≥ − . Then Φ has the SNEP. If,moreover, Φ( x ) is C n for every x ∈ X and n ≥ − , then Φ has the SEP. A space S is called C ω , or aspherical , if every continuous map g : S k → S , k ≥ −
1, can be extended to a continuous map h : B k +1 → S . We shall saythat a family S ⊂ F ( Y ) of subsets of a metric space ( Y, ρ ) is uniformly equi- LC ω if for every ε > δ ( ε ) > S ∈ S , everycontinuous map of the k -sphere ( k ≥ −
1) in S of diameter < δ ( ε ) can be extendedto continuous map of the ( k + 1)-ball into a subset of S of diameter < ε .In view of Theorems 6.3 and 6.4, the following question seems natural. Question 21.
Let X be a strongly countable-dimensional paracompact space,( Y, ρ ) be a complete metric space, and Φ : X → F ( Y ) be an l.s.c. mapping suchthat { Φ( x ) : x ∈ X } is uniformly equi- LC ω and each Φ( x ), x ∈ X , is C ω . Doesthere exist a continuous selection for Φ?Suppose that X , Y and Φ are as in Question 21. Then X = S ∞ n =0 F n , for someincreasing sequence of closed sets F n ⊂ X with dim( F n ) ≤ n . Thus, inductively,using Theorem 5.1, one can construct a selection f : X → Y for Φ such that f ↾ F n is continuous, for every n ≥
0. However, the challenge presented in this questionis to make the construction so that the resulting f will be also continuous. C -spaces. A space X has property C , or X is a C -space ,if for any sequence { U n : n ∈ N } of open covers of X there exists a sequence { V n : n ∈ N } of open pairwise-disjoint families in X such that each V n refines U n and S n ∈ N V n is a cover of X . The C -space property was originally defined byW. Haver [46] for compact metric spaces, subsequently Addis and Gresham [1]reformulated Haver’s definition for arbitrary spaces. It should be remarked that a C -space X is paracompact if and only if it is countably paracompact and normal,see e.g. [26, Proposition 1.3]. Every finite-dimensional paracompact space, as wellas every countable-dimensional metrizable space, is a C -space [1], but there existsa compact metric C -space which is not countable-dimensional [73]. ELECTIONS AND HIGHER SEPARATION AXIOMS 31
A set-valued mapping Φ : X Y is called lower locally constant [37] if the set { x ∈ X : K ⊂ Φ( x ) } is open in X , for every compact subset K ⊂ Y . Thisproperty appeared in a paper of Uspenskij [96]; later on, it was used by someauthors (see, for instance, [10, 97]) under the name “strongly l.s.c.”, while inpapers of other authors strongly l.s.c. was already used for a different property ofset-valued mappings (see, for instance, [32]). Clearly, every lower locally constantmapping is l.s.c. but the converse fails in general and counterexamples abound.In fact, if we consider a single-valued map f : X → Y as a set-valued one, then f is l.s.c. if and only if it is continuous, while f will be lower locally constant ifand only if it is locally constant. Thus, the term “lower locally constant” providessome natural analogy with the single-valued case.The following theorem was obtained by Uspenskij [96]. Theorem 6.5 ([96]) . Let X be a paracompact C -space, Y be a topological space,and Φ : X Y be a lower locally constant mapping with aspherical values. Then Φ has a continuous selection. In view of the method for proving Theorem 5.1, the above selection theoremcan be considered as a “selection approximation property” for the case of l.s.c.mappings, see Proposition 2.11. This brings the following natural question.
Question 22.
Let X be a paracompact C -space, ( Y, ρ ) be a complete metricspace and Φ : X → F ( Y ) be an l.s.c. mapping. What conditions on the family { Φ( x ) : x ∈ X } will guarantee the existence of a continuous selection for Φ? Whatif { Φ( x ) : x ∈ X } is uniformly equi- LC ω and each Φ( x ), x ∈ X , is C ω ?Turning to stronger properties in the setting of this question, let us recall thata metrizable space S is an Absolute Extensor for the metrizable spaces, or shortlyan AE , if every continuous map from a closed subset A of a metrizable space X into S can be extended to a continuous map of X into S . If every continuousmap from a closed subset A of a metrizable space X into S can be extended toa continuous map in S over some neighbourhood of A in X , then S is called an ANE . A collection S of subsets of a metric space ( Y, ρ ) is called uniformly equi-
LAE ( Local Absolute Extensor ) [72] if for every ε > δ > g is a continuous map from a closed subset A of a metrizable space X intoany S ∈ S with diam( g ( A )) < δ , then it has a continuous extension f : X → S with diam( f ( X )) < ε . A collection S of subsets of a metric space ( Y, ρ ) is called uniformly equi- LC ( Locally Contractible ) [72] if for every ε > δ > p ∈ S ∈ S , then O δ ( p ) ∩ S is contractible over a subset of S ofdiameter < ε . Clearly, S is an AE implies that S is C ω , and each uniformlyequi- LAE family S is is equi- LC ω . Moreover, the following was shown by Pixley[72, Theorem 3.1]. Theorem 6.6 ([72]) . For a collection S of subsets of a metric space ( Y, ρ ) , thefollowing conditions are equivalent :(a) The collection S is uniformly equi- LAE . (b) Each S ∈ S is an ANE and S is uniformly equi- LC . Regarding the role of such properties in the selection problem for l.s.c. mappings,the following example was given by Pixley [72, Theorem 1.1].
Theorem 6.7 ([72]) . Let Q = [0 , ω be the Hilbert cube. Then there exists anl.s.c. mapping Φ : Q → F ( Q ) such that (i) The collection { Φ( x ) : x ∈ Q } is uniformly equi- LC , (ii) Each Φ( x ) , x ∈ Q , is either a point, or homeomorphic to a k -cell ( forsome k ≥ , or homeomorphic to Q , (iii) There is no continuous selection for Φ . In fact, Φ has no the SNEP atsome singleton of Q . Uniformly equi-
LAE families are another candidate to try Question 22.
Question 23.
Let X be a paracompact C -space, ( Y, ρ ) be a complete metricspace, and Φ : X → F ( Y ) be an l.s.c. mapping such that { Φ( x ) : x ∈ X } isuniformly equi- LAE and each Φ( x ), x ∈ X , is AE . Does there exist a continuousselection for Φ?Let | Σ | be the geometric realisation of a simplicial complex Σ, and V Σ be itsvertices. As a topological space, we will consider | Σ | endowed with the Whiteheadtopology . In this topology, a subset U ⊂ | Σ | is open if and only if U ∩ | σ | isopen in | σ | , for every simplex σ ∈ Σ. Motivated by the Lefschetz characterisation[52] of compact metrizable absolute neighbourhood retracts, Pixley considered thefollowing stronger condition. A collection S of subsets of a metric space ( Y, ρ ) iscalled uniformly equi- ( L ) if for each ε >
0, there exists δ > K ⊂ Σ with V Σ ⊂ K , eachcontinuous g : | K | → S with diam( g ( | K ∩ σ | )) < δ for every σ ∈ Σ, can beextended to a continuous f : | Σ | → S such that diam( f ( | σ | )) < ε , σ ∈ Σ. Pixleyremarked that each uniformly equi-( L ) family is uniformly equi- LAE based on aresult of Lefschetz [52, (6.6)] and the extension theorem of Dugundji [21, Theorem3.1], but the converse is not true due to Theorem 6.7. Thus, he stated implicitlythe following question.
Question 24 ([72]) . Let X be a paracompact space, ( Y, ρ ) be a complete metricspace, and Φ : X → F ( Y ) be an l.s.c. mapping such that { Φ( x ) : x ∈ X } isuniformly equi-( L ) and each Φ( x ), x ∈ X , is an AE. Does there exist a continuousselection for Φ? What if Φ is ρ -continuous? ELECTIONS AND HIGHER SEPARATION AXIOMS 33
For spaces S and Y , we use C ( S, Y ) to denotethe set of all continuous maps from S to Y endowed with the compact-opentopology. In fact, C ( S, Y ) will be used in the case of a compact S and a metrizable Y , where this topology is the uniform topology on C ( S, Y ) generated by anycompatible metric on Y . Finally, let Q = [0 , ω be the Hilbert cube; I n = [0 , n be the n -cube, and I to be the 0-cube (i.e. a singleton).A closed set A of a metrizable space Y is said to be a Z n -set in Y , where n ≥ C ( I n , Y \ A ) is dense in C ( I n , Y ), and A is called a Z -set in Y if C ( Q , Y \ A ) isdense in C ( Q , Y ), see [9] and [93]. The collection of all Z n -sets ( Z -sets) in Y willbe denoted by Z n ( Y ) (respectively, Z ( Y )). Let us remark that the elements of Z ( Y ) are precisely the closed nowhere dense subsets of Y . Moreover, it is wellknown that Z ( Y ) = T n ≥ Z n ( Y ), see e.g. [6, Proposition 2.1 in § Z -sets are sometimes called Z ∞ -sets. Finally, let us alsoremark that a closed subset A ⊂ Q is a Z -set in Q iff the identity map of Q can be uniformly approximated by continuous self-maps of Q with range entirelycontained Q \ A , see e.g. [62, Lemma 6.2.3].The following characterisation of C -spaces was obtained by Uspenskij [96, The-orem 1.4]. Theorem 6.8 ([96]) . A compact space X is a C -space if and only if for everycontinuous mapping Ψ : X → Z ( Q ) there exists a continuous map f : X → Q such that f ( x ) / ∈ Ψ( x ) , for all x ∈ X . Subsequently, Theorem 6.8 was extended to all paracompact C -spaces in [43,Theorem 1.1]. Theorem 6.9 ([43]) . A paracompact space X is a C -space if and only if whenever E is a Banach space, Φ : X → F c ( E ) is l.s.c. and Ψ n : X → F ( E ) , n ∈ N , areclosed-graph mappings with Ψ n ( x ) ∩ Φ( x ) ∈ Z (Φ( x )) , for all x ∈ X and n ∈ N ,there exists a continuous map f : X → E with f ( x ) ∈ Φ( x ) \ S n ∈ N Ψ n ( x ) , forevery x ∈ X . Every continuous mapping Ψ : X → Z ( Q ) has a closed graph. Hence, bytaking Φ( x ) = Q , x ∈ X , Theorem 6.8 follows from Theorem 6.9. The interestedreader is referred to [97], where finite C -spaces were characterised in a similarmanner; also to [44], where a natural finite-dimensional version of Theorem 6.9(in terms of Z n -sets) was obtained. G δ -Sets. If Y is completely metrizable and anabsolute extensor for the metrizable spaces, then it is also an absolute extensorfor the collectionwise normal spaces [17]. In particular, this is true for everyconvex G δ -subset Y of a Banach space E . Indeed, Y is an absolute extensorfor the metrizable spaces being convex (by Dugundji’s extension theorem [21]), and is also completely metrizable being a G δ -subset of a complete metric space.Motivated by this, the following problem was stated by Michael in [61, Problem396], it became known as Michael’s G δ -problem . Question 25 ([61]) . Let E be a Banach space, Y ⊂ E be a convex G δ -subset of E , X be a paracompact space and Φ : X → F c ( Y ) be an l.s.c. mapping. Does Φhave a continuous selection?In general, the answer to this question is in the negative due to a counterexampleconstructed by Filippov [27, 28], see also [78]. However, Question 25 was alsoresolved in the affirmative in a number of partial cases. As Michael remarked in[61, Remark 3.11], the answer is “Yes” if conv( K ) ⊂ Φ( x ) for every compact subset K ⊂ Φ( x ) and x ∈ X . For instance, this is true if Y is a countable intersection ofopen convex sets, or dim Φ( x ) < ∞ , for all x ∈ X . Various related observationsfor this special case can be found in [29, 68]. Another remark made by Michael isthat the answer is “Yes” provided dim( X ) < ∞ [61, Remark 3.6].All these special cases were generalised in [31, Theorem 1.1] by showing that inthe setting of Question 25, continuous selections are equivalent to the existenceof convex-valued usco selections (i.e. convex-compact-valued u.s.c. selections). Theorem 6.10 ([31]) . For a paracompact space X and a G δ -subset Y of a Banachspace E , the following are equivalent :(a) Every l.s.c.
Φ : X → F c ( Y ) has a continuous selection. (b) Every l.s.c.
Φ : X → F c ( Y ) has an usco convex-valued selection. Furthermore, Theorem 6.10 implies that the answer is “Yes” when X is acountable-dimensional metrizable space [31, Corollary 1.2], or when it is a stronglycountable-dimensional paracompact space [31, Corollary 1.3].Finally, let us explicitly remark that Question 25 was also resolved in the affir-mative in the realm of C -spaces [43, Theorem 4.4]. Theorem 6.11 ([43]) . Let X be a paracompact C -space, E be a Banach spaceand Y be a G δ -subset of E . Then every l.s.c. mapping Φ : X → F c ( Y ) has acontinuous selection. This result is, in fact, based on Theorem 6.9 and the following property ofconvex G δ -sets in Banach space, see [43, Lemma 4.3]. Lemma 6.12 ([43]) . If E is a Banach space, H ⊂ E is a convex G δ -subset of E and F ⊂ E is a closed set with F ∩ H ∈ Z ( H ) , then F ∩ H ∈ Z (cid:0) H (cid:1) . Motivated by this, the following question was posed by Repovˇs and Semenovin [77, Problem 1.6], and subsequently in [79, Problem 2.6].
ELECTIONS AND HIGHER SEPARATION AXIOMS 35
Question 26 ([77, 79]) . Let X be a paracompact space such that for every G δ -subset Y ⊂ E of a Banach space E , every l.s.c. Φ : X → F c ( Y ) has a continuousselection. Does this imply that X is a C -space?Question 26 was resolved by Karassev [48, Theorem 4.6] for the case of weaklyinfinite-dimensional compact spaces. Theorem 6.13 ([48]) . Let X be a compact space such that for every G δ -subset Y ⊂ E of a Banach space E , every l.s.c. mapping Φ : X → F c ( Y ) has a contin-uous selection. Then X is weakly infinite-dimensional. As remarked in [79], perhaps what is also interesting is the implicit relationof Theorem 6.13 to one of the main problems in infinite dimension theory ofwhether every weakly infinite-dimensional compact metric space has property C .In the realm of compact spaces, there are various characterisations of weak infinite-dimensionality. In case of Theorem 6.13, Karassev used the following property,compare with Theorem 6.8. Theorem 6.14 ([81]) . A compact space X is weakly infinite-dimensional if andonly if for every continuous map g : X → Q in the Hilbert cube Q , there exists acontinuous map f : X → Q with f ( x ) = g ( x ) , for all x ∈ X . Another aspect of Question 25 was considered in [39]. Namely, in view of therelationship between selections and extension (see Theorem 2.6), the followingquestion was posed in [39, Question 2].
Question 27 ([39]) . Let X be a collectionwise normal space, E be a Banachspace, Y ⊂ E be a convex G δ -subset of E and Φ : X → C ′ c ( Y ) be an l.s.c.mapping. Does Φ have a continuous selection?This question is not only similar to Question 25, but most of the affirmativesolutions of Question 25 remain valid for it as well. Indeed, if conv( K ) ⊂ Φ( x )for every compact subset K ⊂ Φ( x ) and x ∈ X , by a result of [15], Φ has anl.s.c. selection ϕ : X → C c ( Y ). Hence, Φ also has a continuous selection because,by Theorem 2.3, so does ϕ . If X is finite-dimensional, the answer is also “Yes”,and follows directly from a selection theorem in [30]. The answer to Question 27is also “Yes” if X is strongly countable-dimensional. In this case, the mappingΦ : X → C ′ c ( Y ) has an l.s.c. weak-factorisation ( Z, g, ϕ ) with Z being a stronglycountable-dimensional space, see, for instance, the proof of [69, Theorem 5.3].Then just as in the proof of Proposition 3.2, Φ has a continuous selection becauseso does the mapping conv[ ϕ ] Y : Z → F c ( Y ), where the closure is in Y .In contrast to Theorem 6.11, Question 27 is still open for collectionwise normal C -spaces. Question 28.
Let X be a collectionwise normal C -space, E be a Banach space, Y be a convex G δ -subset of E and Φ : X → C ′ c ( Y ) be an l.s.c. mapping. Doesthere exist a continuous selection for Φ?As mentioned before, a countably paracompact normal C -space is paracompact,in which case the answer is “Yes”, by Theorem 6.11. Hence, Question 28 is forcollectionwise normal spaces which are not countably paracompact.Every countable-dimensional metrizable space has property C , and more gener-ally, every countable-dimensional hereditarily paracompact space is a C -space [1,Corollary 2.10]. However, it seems it is still an open question if every countable-dimensional paracompact space is a C -space [26, Question 1]. This brings thefollowing related question in terms of selections and G δ -sets of Banach spaces. Question 29.
Let X be a countable-dimensional paracompact space, Y ⊂ E bea (convex) G δ -subset of a Banach space and Φ : X → F c ( Y ) be an l.s.c. mapping.Does there exist a continuous selection for Φ? There is a further relationship between C -spaces and the property of weakly infinite-dimensional spaces used by Karassev[48], see Theorem 6.14. A pair of set-valued mappings ϕ, ψ : X Y are called disjoint if ϕ ( x ) ∩ ψ ( x ) = ∅ , for every x ∈ X . It was shown by Dranishnikov [20,Theorem 1], see also [19], that the fibration η = ∞ Y i =0 ν i : ∞ Y i =0 S (2 i ) → RP (2 i ) does not accept two disjoint usco sections, where ν k : S k → RP k is a 2-fold coveringmap of the k -sphere onto the real projective k -space. Here, an usco section for η is meant an usco set-valued selection for the inverse set-valued mapping η − .Since η is open, its inverse is l.s.c., so this is an example of an l.s.c. infinite-valuedmapping which doesn’t admit a pair of disjoint usco selections. On the otherhand, the following was shown in [45, Corollary 4.4]. Theorem 6.15 ([45]) . Let X be a paracompact C -space, Y be completely metriz-able and Φ : X → F ( Y ) be an l.s.c. mapping such that each Φ( x ) , x ∈ X , isinfinite. Then Φ has a pair of disjoint usco selections. Regarding disjoint usco selections, the following part of [49, Problem 1516] isstill open.
Question 30 ([49]) . Let X be a metrizable space such that for every metrizablespace Y , any l.s.c. mapping Φ : X → C ( Y ) with perfect point-images Φ( x ), x ∈ X , admits disjoint u.s.c. selections. Is it true that X is a C -space?A map f : X → E is said to avoid some set Z ⊂ E , if f ( x ) / ∈ Z for all x ∈ X .In case E is a linear space and Z = { } is the singleton of the origin of E , the ELECTIONS AND HIGHER SEPARATION AXIOMS 37 map is simply called -avoiding . In [60], Michael considered the following naturalproblem for -avoiding selections. Question 31 ([60]) . Let X be a paracompact space, E be a Banach space andΦ : X → F c ( E ) be an l.s.c. mapping such that Φ( x ) = { } , for every x ∈ X .Under what conditions, does Φ have a continuous -avoiding selection?He remarked that in setting of selection theorems such as Theorem 2.4, theconstructed continuous selections cannot be chosen to be -avoiding (even whenΦ( x ) = { } for all x ∈ X ), and provided several examples, see [60, Examples 10.1and 10.2]. In case of dimension restrictions on X , or strengthening the continuityof Φ, he obtained the following theorems. Theorem 6.16 ([60]) . Let X be a paracompact space, E be a Banach space and Φ : X → F c ( E ) be an l.s.c. mapping such that dim( X ) < dim Φ( x ) , whenever x ∈ X and ∈ Φ( x ) . Then Φ has a continuous -avoiding selection. Theorem 6.17 ([60]) . Let X be an arbitrary space, E be a Banach space and Φ : X → F c ( E ) be a norm-continuous mapping with dim Φ( x ) = ∞ , for all x ∈ X .If u k u k ∈ Φ( x ) , whenever = u ∈ Φ( x ) and x ∈ X , then Φ has a continuous -avoiding selection. Regarding the proper place of Theorem 6.17, Michael stated the following ques-tion in [61, Problem 395].
Question 32 ([61]) . Let X be a paracompact space, E be an infinite-dimensionalBanach space and Φ : X → F ( E ) be an l.s.c. mapping with each Φ( x ) a linearsubspace of deficiency one (or of finite deficiency) in E . Must Φ have a continuous -avoiding selection?According to [61, Remark 3.7], the answer to this question is “No” if it is onlyassumed that each Φ( x ) is infinite-dimensional. This follows from the mentionedexample of Dranishnikov [19] and a similar example of Toru´nczyk and West [94].Here are some further remarks regarding Question 32. Corollary 6.18.
Let X be a paracompact C -space, E be a Banach space and Φ : X → F c ( E ) be an l.s.c. mapping with dim Φ( x ) = ∞ , for every x ∈ X . Then Φ has a continuous -avoiding selection.Proof. Whenever x ∈ X , the singleton { } is a Z -set in Φ( x ) because Φ( x ) isinfinite-dimensional; equivalently, { } a Z n -set in Φ( x ) for every n ≥ Z n -set). Then theexistence of a continuous -avoiding selection for Φ follows from Theorem 6.9 bytaking Ψ n ( x ) = { } for all x ∈ X and n ∈ N . (cid:3) As Michael emphasised in [60], the benefit of Theorems 6.16 and 6.17 is thatthey actually show the existence of continuous selections avoiding given continuous maps. Namely, suppose that g : X → E is a continuous map in a Banach space,and Φ : X → F c ( E ) is an l.s.c. mapping. Then one can consider the l.s.c. mappingΦ − g : X → F c ( E ), defined by [Φ − g ]( x ) = Φ( x ) − g ( x ), x ∈ X . If Φ − g hasa continuous -avoiding selection f : X → E , then h = f + g : X → E is acontinuous selection for Φ with h ( x ) = g ( x ) for all x ∈ X . Based on this and thecharacterisation of weakly infinite-dimensional compact spaces in Theorem 6.14,we also have the following observation. Proposition 6.19.
Let X be a compact space such that for every ( separable ) Banach space E , every l.s.c. mapping Φ : X → F c ( E ) with dim Φ( x ) = ∞ , forall x ∈ X , has a continuous -avoiding selection. Then X is weakly infinite-dimensional.Proof. Consider the Hilbert cube Q as a compact convex subset of E = ℓ = ℓ ( N ),or, alternatively, of ℓ = ℓ ( N ). Take a continuous map g : X → Q , and nextdefine an l.s.c. mapping Φ : X → F c ( E ) by Φ( x ) = Q − g ( x ), x ∈ X , see [60,Example 10.2]. Then dim Φ( x ) = ∞ , for all x ∈ X , and by condition, Φ has acontinuous -avoiding selection h : X → E . As discussed above, this implies that f = h + g : X → Q is a continuous map with f ( x ) = g ( x ), for every x ∈ X .According to Theorem 6.14, X is weakly infinite-dimensional. (cid:3) Acknowledgement.
The author is very much indebted to the referee who con-tributed an alternative proof of Theorem 4.2, and whose helpful and essentialcorrections greatly improved the final version of this paper.
References [1] D. Addis and J. Gresham,
A class of infinite-dimensional spaces. I. Dimension theory andAlexandroff ’s problem , Fund. Math. (1978), no. 3, 195–205.[2] S. M. Ageev and D. Repovsh,
A unified finite-dimensional selection theorem , Sibirsk. Mat.Zh. (1998), no. 5, 971–981, i, (in Russian); English translation in Siberian Math. J. (1998), no. 5, 835-843.[3] R. A. Al´o and L. I. Sennott, Extending linear space-valued functions , Math. Ann. (1971), 79–86.[4] R. Arens,
Extension of coverings, of pseudometrics, and linear-space-valued mappings ,Canad. J. Math. (1953), 211–215.[5] J.-P. Aubin and H. Frankowska, Set-Valued Analysis , System & Control, Foundation andApplications, vol. 2, Birkh¨auser, Boston, 1990.[6] C. Bessaga and A. Pe lczy´nski,
Selected topics in infinite-dimensional topology , PWN, War-saw, 1975.[7] R. H. Bing,
Metrization of topological spaces , Canad. J. Math. (1951), 175–186.[8] K. Borsuk, Sur les prolongements des transformations continues , Fund. Math. (1936),99–110.[9] A. Chigogidze, Inverse Spectra , North-Holland, Amsterdam, 1996.[10] A. Chigogidze and V. Valov,
Extensional dimension and C -spaces , Bull. London Math. Soc. (2002), no. 6, 708–716. ELECTIONS AND HIGHER SEPARATION AXIOMS 39 [11] M. Choban,
Many-valued mappings and Borel sets , Trudy Moskov. Mat. Obˇsˇc. (1970),229–250, (in Russian); English translation in Trans. Moscow Math. Soc. (1970), 258–280.[12] , Many-valued mappings and Borel sets. II , Trudy Moskov. Mat. Obˇsˇc. (1970),277–301, (in Russian); English translation in Trans. Moscow Math. Soc. (1970), 286–310.[13] M. Choban and S. Nedev, Factorization theorems for set-valued mappings, set-valued selec-tions and topological dimension , Math. Balkanica (1974), 457–460, (in Russian).[14] , Continuous selections for mappings with generalized ordered domain , Math.Balkanica (N.S.) (1997), no. 1-2, 87–95.[15] M. Choban and V. Valov, On a theorem of E. Michael on selections , C. R. Acad. BulgareSci. (1975), 871–873, (in Russian).[16] C. H. Dowker, On countably paracompact spaces , Canad. J. of Math. (1951), 291–224.[17] , On a theorem of Hanner , Ark. Mat. (1952), 307–313.[18] , Local dimension of normal spaces , Quart. J. of Math. Oxford (1955), 101–120.[19] A. Dranishnikov, Q -bundles without disjoint sections , Funktsional. Anal. i Prilozhen. (1988), no. 2, 79–80, (in Russian); English translation in Funct. Anal. Appl. (1988),no. 2, 151–152.[20] , A fibration that does not accept two disjoint many-valued sections , Topology Appl. (1990), 71–73.[21] J. Dugundji, An extension of Tietze’s theorem , Pacific J. Math. (1951), 353–367.[22] J. Dugundji and E. Michael, On local and uniformly local topological properties , Proc. Amer.Math. Soc. (1956), 304–307.[23] J. Dydak, Extension theory: the interface between set-theoretic and algebraic topology ,Topology Appl. (1996), no. 1-3, 225–258.[24] R. Engelking, General topology, revised and completed edition , Heldermann Verlag, Berlin,1989.[25] R. Engelking, R. W. Heath, and E. Michael,
Topological well-ordering and continuous se-lections , Invent. Math. (1968), 150–158.[26] V. V. Fedorchuk, Weakly infinite-dimensional spaces , Uspekhi Mat. Nauk (2007),no. 2(374), 109–164 (in Russian); English translation in Russian Math. Surveys (2007),no. 2, 323–374.[27] V. V. Filippov, On a question of E. A. Michael , Comment. Math. Univ. Carolin. (2004),no. 4, 735–737.[28] , On a problem of E. Michael , Mat. Zametki (2005), no. 4, 638–640, (in Russian);English translation in Math. Notes (2005), no. 3-4, 597–599.[29] V. L. Ge˘ınts and V. V. Filippov, On convex hulls of compact sets of probability measures withcountable supports , Funktsional. Anal. i Prilozhen. (2011), no. 1, 83–88, (in Russian);English translation in Funct. Anal. Appl. (2011), no. 1, 69–72.[30] V. Gutev, Continuous selections and finite-dimensional sets in collectionwise normal spaces ,C. R. Acad. Bulgare Sci. (1986), no. 5, 9–12.[31] , Continuous selections, G δ -subsets of Banach spaces and usco mappings , Comment.Math. Univ. Carolin. (1994), no. 3, 533–538.[32] , Factorizations of set-valued mappings with separable range , Comment. Math. Univ.Carolin. (1996), no. 4, 809–814.[33] , Extending of homotopies via selections and extending of selections via homotopies ,Proceedings of the Eighth Prague Topological Symposium (1996), Topol. Atlas, North Bay,ON, 1997, pp. 128–148 (electronic).[34] ,
An exponential mapping over set-valued mappings , Houston J. Math. (2000),no. 4, 721–739. [35] , Weak factorizations of continuous set-valued mappings , Topology Appl. (2000),33–51.[36] ,
Extending of selections via homotopies , JP J. Geom. Topol. (2002), no. 2, 117–140.[37] , Selections and approximations in finite-dimensional spaces , Topology Appl. (2005), 353–383.[38] ,
Open relations and collectionwise normality , Bull. Pol. Acad. Sci. Math. (2017),no. 1, 69–79.[39] V. Gutev and N. R. Loufouma Makala, Selections, extensions and collectionwise normality ,J. Math. Anal. Appl. (2010), no. 2, 573–577.[40] V. Gutev and S. Nedev,
Continuous selections and reflexive Banach spaces , Proc. Amer.Math. Soc. (2001), no. 6, 1853–1860.[41] V. Gutev, H. Ohta, and K. Yamazaki,
Selections and sandwich-like properties via semi-continuous Banach-valued functions , J. Math. Soc. Japan (2003), no. 2, 499–521.[42] , Extensions by means of expansions and selections , Set-Valued Analysis (2006),69–104.[43] V. Gutev and V. Valov, Continuous selections and C -spaces , Proc. Amer. Math. Soc. (2002), 233–242.[44] , Dense families of selections and finite-dimensional spaces , Set-Valued Analysis (2003), 373–391.[45] , Open maps having the Bula property , Fund. Math. (2009), no. 2, 91–104.[46] W. Haver,
A covering property for metric spaces , Topology Conference (Virginia Polytech.Inst. and State Univ., Blacksburg, Va., 1973), Springer, Berlin, 1974, pp. 108–113. LectureNotes in Math., Vol. 375.[47] T. Kandˆo,
Characterization of topological spaces by some continuous functions , J. Math.Soc. Japan (1954), 45–54.[48] A. Karassev, Michael’s problem and weakly infinite-dimensional spaces , Topology Appl. (2008), 1694–1698.[49] A. Karassev, M. Tuncali, and V. Valov,
Topology in North Bay: Some Problems in Contin-uum Theory, Dimension Theory and Selections , Open Problems in Topology 2 (E. Pearl,ed.), Elsevier BV., Amsterdam, 2007, pp. 697–710.[50] M. Katˇetov,
On real-valued functions in topological spaces , Fund. Math. (1951), 85–91.[51] , Correction to “On real-valued functions in topological spaces” (Fund. Math. 38(1951), pp. 85–91) , Fund. Math. (1953), 203–205.[52] S. Lefschetz, Topics in Topology , Annals of Mathematics Studies, no. 10, Princeton Univer-sity Press, Princeton, N. J., 1942.[53] E. Michael,
Point-finite and locally finite coverings , Canad. J. Math. (1955), 275–279.[54] , Continuous selections I , Ann. of Math. (1956), 361–382.[55] , Continuous selections II , Ann. of Math. (1956), 562–580.[56] , Continuous selections III , Ann. of Math. (1957), 375–390.[57] , A theorem on semi-continuous set-valued functions , Duke Math. J (1959), 647–651.[58] , Uniform AR’s and ANR’s , Compositio Math. (1979), no. 2, 129–139.[59] , Continuous selections and finite-dimensional sets , Pacific J. Math. (1980), 189–197.[60] , Continuous selections avoiding a set , Topology Appl. (1988), 195–213.[61] , Some problems , Open Problems in Topology (J. van Mill and G. M. Reed, eds.),North-Holland, Amsterdam, 1990, pp. 273–278.
ELECTIONS AND HIGHER SEPARATION AXIOMS 41 [62] J. v. Mill,
Infinite Dimensional Topology , Prerequisites and Introduction, North-Holland,Amsterdam, 1989.[63] K. Morita,
Star-finite coverings and star-finite property , Math. Japonicae (1948), 60–68.[64] , Paracompactness and product spaces , Fund. Math. (1961/1962), 223–236.[65] , Products of normal spaces with metric spaces , Math. Ann. (1964), 365–382.[66] , ˇCech cohomology and covering dimension for topological spaces , Fund. Math. (1975), no. 1, 31–52.[67] J. Nagata, On the countable sum of zero-dimensional metric spaces , Fund. Math. (1959/1960), 1–14.[68] I. Namioka and E. Michael, A note on convex G δ -subsets of Banach spaces , Topology Appl. (2008), no. 8, 858–860.[69] S. Nedev, Selection and factorization theorems for set-valued mappings , Serdica (1980),291–317.[70] , A selection example , C. R. Acad. Bulgare Sci. (1987), no. 11, 13–14.[71] S. Nedev and V. Valov, Normal selectors for the normal spaces , C. R. Acad. Bulgare Sci. (1984), no. 7, 843–846.[72] C. P. Pixley, An example concerning continuous selections of infinite-dimensional spaces ,Proc. Amer. Math. Soc. (1974), 237–244.[73] R. Pol, A weakly infinite-dimensional compactum which is not countable-dimensional , Proc.Amer. Math. Soc. (1981), no. 4, 634–636.[74] T. Przymusi´nski, Collectionwise normality and absolute retracts , Fund. Math. (1978),61–73.[75] , Collectionwise normality and extensions of continuous functions , Fund. Math. (1978), 75–81.[76] T. C. Przymusi´nski and M. L. Wage, Collectionwise normality and extensions of locallyfinite coverings , Fund. Math. (1980), no. 3, 175–187.[77] D. Repovˇs and P. V. Semenov,
Continuous selections of multivalued mappings , RecentProgress in General Topology, II, North-Holland, Amsterdam, 2002, pp. 423–461.[78] ,
On closedness assumptions in selection theorems , Topology Appl. (2007),no. 10, 2185–2195.[79] ,
Continuous selections of multivalued mappings , Recent Progress in General Topol-ogy. III, Atlantis Press, Paris, 2014, pp. 711–749.[80] V. ˇSediv´a,
On collectionwise normal and hypocompact space , Czechoslovak Math. J. (1959), 50–62, (in Russian).[81] J. Segal and T. Watanabe,
Universal maps and infinite-dimensional spaces , Bull. Pol. Acad.Sci. Math. (1991), no. 3-4, 225–228.[82] H. L. Shapiro, Extensions of pseudometrics , Canad. J. Math. (1966), 981–998.[83] I. Shishkov, Selections of lower semi-continuous mappings into Hilbert spaces , C. R. Acad.Bulgare Sci. (2000), no. 7, 5–8.[84] , Σ -products and selections of set-valued mappings , Comment. Math. Univ. Carolin. (2001), no. 1, 203–207.[85] , On a conjecture of Choban, Gutev and Nedev , Math. Balkanica (N.S.) (2004),no. 1-2, 193–196.[86] , Selections of set-valued mappings with hereditarily collectionwise normal domain ,Topology Appl. (2004), no. 1-3, 95–100.[87] , c -paracompactness and selections , Math. Balkanica (N.S.) (2007), no. 1-2, 51–57.[88] , Collectionwise normality and selections into Hilbert spaces , Topology Appl. (2008), no. 8, 889–897. [89] J. C. Smith,
Properties of expandable spaces , General Topology and its Relations to ModernAnalysis and Algebra, III (Proc. Third Prague Topological Sympos., 1971), Academia,Prague, 1972, pp. 405–410.[90] ,
Collectionwise normality and expandability in quasi- k -spaces , Topology, Vol. II(Proc. Fourth Colloq., Budapest, 1978), Colloq. Math. Soc. J´anos Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 1111–1125.[91] H. Tong, Some characterizations of normal and perfectly normal spaces , Bull. Amer. Math.Soc. (1948), no. 1, 65.[92] , Some characterizations of normal and perfectly normal spaces , Duke Math. J. (1952), 289–292.[93] H. Toru´nczyk, Concerning locally homotopy negligible sets and characterization of l -manifolds , Fund. Math. (1978), 93–110.[94] H. Toru´nczyk and J. West, Fibrations and bundles with Hilbert cube manifold fibers , Mem.Amer. Math. Soc. (1989), no. 406, iv+75.[95] S. L. Troyanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces , Studia Math. (1971), 173–180.[96] V. Uspenskij, A selection theorem for C -spaces , Topology Appl. (1998), 351–374.[97] V. Valov, Continuous selections and finite C -spaces , Set-Valued Analysis (2002), no. 1,37–51.[98] V. ˇSmulian, On the principle of inclusion in the space of the type (B), Rec. Math. [Mat.Sbornik] N.S. (1939), 317–328, (in Russian).[99] B. R. Wenner,
Finite-dimensional properties of infinite-dimensional spaces , Pacific J. Math. (1972), 267–276. Department of Mathematics, Faculty of Science, University of Malta, MsidaMSD 2080, Malta
E-mail address ::