Constructing Selections Stepwise Over Cones of Simplicial Complexes
aa r X i v : . [ m a t h . GN ] J un CONSTRUCTING SELECTIONS STEPWISE OVER CONES OFSIMPLICIAL COMPLEXES
VALENTIN GUTEV
Abstract.
It is obtained a natural generalisation of Uspenskij’s selection char-acterisation of paracompact C -spaces. The method developed to achieve thisresult is also applied to give a simplified proof of a similar characterisation ofparacompact finite C -space obtained previously by Valov. Another applicationis a characterisation of finite-dimensional paracompact spaces which generalisesboth a remark done by Michael and a result obtained by the author.
1. Introduction
All spaces in this paper are Hausdorff topological spaces. A space X has prop-erty C , or is a C -space , if for any sequence { U n : n < ω } of open covers of X there exists a sequence { V n : n < ω } of open pairwise-disjoint families in X suchthat each V n refines U n and S n<ω V n is a cover of X . The C -space propertywas originally defined by W. Haver [10] for compact metric spaces, subsequentlyAddis and Gresham [1] reformulated Haver’s definition for arbitrary spaces. Itshould be remarked that a C -space X is paracompact if and only if it is countablyparacompact and normal, see e.g. [6, Proposition 1.3]. Every finite-dimensionalparacompact space, as well as every countable-dimensional metrizable space, is a C -space [1], but there exists a compact metric C -space which is not countable-dimensional [14].In what follows, we will use Φ : X Y to designate that Φ is a map from X to the nonempty subsets of Y , i.e. a set-valued mapping . A map f : X → Y is a selection for Φ : X Y if f ( x ) ∈ Φ( x ), for all x ∈ X . A mapping Φ : X Y is lower locally constant , see [8], if the set { x ∈ X : K ⊂ Φ( x ) } is open in X , forevery compact subset K ⊂ Y . This property appeared in a paper of Uspenskij [15];later on, it was used by some authors (see, for instance, [3, 16]) under the name“strongly l.s.c.”, while in papers of other authors strongly l.s.c. was already usedfor a different property of set-valued mappings (see, for instance, [7]). Regarding Date : June 24, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Lower locally constant mapping, continuous selection, connectednessin finite dimension, C -space, finite C -space, finite-dimensional space, simplicial complex, nerve. our terminology, let us remark that a singleton-valued mapping (i.e. a usual map)is lower locally constant precisely when it is locally constant.Finally, let us recall that a space S is aspherical if every continuous map of the k -sphere ( k ≥
0) in S can be extended to a continuous map of the ( k + 1)-ball in S . The following theorem was obtained by Uspenskij [15, Theorem 1.3]. Theorem 1.1 ([15]) . A paracompact space X is a C -space if and only if forevery topological space Y , each lower locally constant aspherical-valued mapping Φ : X Y has a continuous selection. For k ≥ S, B ⊂ Y , we will write that S k ֒ → B if every continuousmap of the k -sphere in S can be extended to a continuous map of the ( k + 1)-ballin B . Similarly, for mappings ϕ, ψ : X Y , we will write ϕ k ֒ → ψ to express that ϕ ( x ) k ֒ → ψ ( x ), for every x ∈ X . In these terms, we shall say that a sequence ofmappings ϕ n : X Y , n < ω , is aspherical if ϕ n n ֒ → ϕ n +1 , for every n < ω . Also,to each sequence of mappings ϕ n : X Y , n < ω , we will associate its union S n<ω ϕ n : X Y , defined pointwise by (cid:2)S n<ω ϕ n (cid:3) ( x ) = S n<ω ϕ n ( x ), x ∈ X .In the present paper, we will show that the mapping Φ : X Y in Theorem 1.1can be replaced by an aspherical sequence of lower locally constant mappings ϕ n : X Y , n < ω . Namely, the following theorem will be proved. Theorem 1.2.
A paracompact space X is a C -space if and only if for everytopological space Y , each aspherical sequence ϕ n : X Y , n < ω , of lower locallyconstant mappings admits a continuous selection for its union S n<ω ϕ n . By taking ϕ n = ϕ n +1 , n < ω , the selection property in Theorem 1.2 immedia-tely implies that of Theorem 1.1. Implicitly, the selection property in Theorem 1.1also implies that of Theorem 1.2 because both these properties are equivalent to X being a C -space. However, the author is not aware of any explicit argument show-ing this. In this regard and in contrast to Theorem 1.1, the proof of Theorem 1.2is straightforward in both directions. Here is briefly the idea behind this proof.In the next section, we deal with a simple construction of continuous extensionsof maps over cones of simplicial complexes, see Proposition 2.1. This construc-tion is applied in Section 4 to special simplicial complexes which are defined inSection 3. Namely, in Section 3, to each cover S n<ω U n of X consisting of fam-ilies U n , n < ω , of subsets of X , we associate a subcomplex ∆( U <ω ) of thenerve N ( U <ω ), where U <ω stands for the disjoint union F n<ω U n , see Exam-ple 3.1. Intuitively, ∆( U <ω ) consists of those simplices σ ∈ N ( U <ω ) which haveat most one vertex in each U n , n < ω . One benefit of this subcomplex is that∆( V <ω ) = N ( V <ω ), whenever the families V n , n < ω , are as in the definingproperty of C -spaces, i.e. pairwise-disjoint, see Proposition 3.2. Another benefit ONSTRUCTING SELECTIONS STEPWISE OVER CONES 3 is that each sequence of covers U n , n < ω , of X generates a natural asphericalsequence of mappings on X . This is done by considering the sequence of sub-complexes ∆( U ≤ n ), n < ω , where each ∆( U ≤ n ) is defined as above with respectto the indexed cover U ≤ n = F nk =0 U k of X . Then each ∆( U ≤ n ), n < ω , gen-erates a simplicial-valued mapping ∆ [ U ≤ n ] : X ∆( U ≤ n ) which assigns to each x ∈ X the simplicial complex ∆ [ U ≤ n ] ( x ) of those simplices σ ∈ ∆( U ≤ n ) for which x ∈ T σ . Finally, we consider the geometric realisation | ∆( U ≤ n ) | of ∆( U ≤ n ) andthe set-valued mapping (cid:12)(cid:12) ∆ [ U ≤ n ] (cid:12)(cid:12) : X | ∆( U ≤ n ) | ⊂ | ∆( U <ω ) | corresponding tothe “geometric realisation” of ∆ [ U ≤ n ] . Thus, for open locally finite covers U n , n < ω , of X , the sequence (cid:12)(cid:12) ∆ [ U ≤ n ] (cid:12)(cid:12) : X | ∆( U <ω ) | , n < ω , is always asphericaland consists of lower locally constant mappings, Propositions 3.3 and 3.4. In Sec-tion 4, by restricting X to be a paracompact space, we show that each asphericalsequence ϕ n : X Y , n < ω , of lower locally constant mappings admits a se-quence F n , n < ω , of closed locally finite interior covers of X and a continuousmap f : | ∆( F <ω ) | → Y such that each composite mapping f ◦ (cid:12)(cid:12) ∆ [ F ≤ n ] (cid:12)(cid:12) : X Y is a set-valued selection for ϕ n , n < ω , see Theorem 4.1. This is applied in Sec-tion 5 to show that the selection problem in Theorem 1.2 is now equivalent to thatof the mappings (cid:12)(cid:12) ∆ [ U ≤ n ] (cid:12)(cid:12) : X | ∆( U <ω ) | , n < ω , corresponding to open locallyfinite covers U n , n < ω , of X , Theorem 5.1. In the same section, this selectionproblem is further reduced to the existence of canonical maps f : X → | ∆( U <ω ) | ,Corollary 5.4. Finally, in Section 6, it is shown that the existence of canonicalmaps f : X → | ∆( U <ω ) | is equivalent to C -like properties of X , Theorem 6.1.Theorem 1.2 is then obtained as a special case of Theorem 6.1, see Corollary 6.2.Here, let us explicitly remark that two other special cases of Theorem 6.1 arecovering two other similar results — Corollary 6.3 (a selection theorem of Valov[16, Theorem 1.1] about finite C -spaces) and Corollary 6.5 (generalising both aremark done by Michael in [15, Remark 2] and a result obtained by the author in[9, Theorem 3.1]).
2. Extensions of maps over cones of simplicial complexes
By a simplicial complex we mean a collection Σ of nonempty finite subsets ofa set S such that τ ∈ Σ, whenever ∅ = τ ⊂ σ ∈ Σ. The set S Σ is the vertex set of Σ, while each element of Σ is called a simplex . The k -skeleton Σ k of Σ ( k ≥ k = { σ ∈ Σ : card( σ ) ≤ k + 1 } , where card( σ ) is thecardinality of σ . In the sequel, for simplicity, we will identify the vertex set of Σwith its 0-skeleton Σ , and will say that Σ is a k -dimensional if Σ = Σ k .The vertex set Σ of each simplicial complex Σ can be embedded as a linearlyindependent subset of some linear normed space. Then to any simplex σ ∈ Σ,we may associate the corresponding geometric simplex | σ | which is the convexhull of σ . Thus, card( σ ) = k + 1 if and only if | σ | is a k -dimensional simplex . VALENTIN GUTEV
Finally, we set | Σ | = S {| σ | : σ ∈ Σ } which is called the geometric realisation of Σ.As a topological space, we will always consider | Σ | endowed with the Whiteheadtopology [17, 18]. This is the topology in which a subset U ⊂ | Σ | is open if andonly if U ∩ | σ | is open in | σ | , for every σ ∈ Σ.The cone Z ∗ v over a space Z with a vertex v is the quotient space of Z × [0 , Z × { } into a single point v . For a simplicialcomplex Σ and a point v with v / ∈ Σ , the cone on Σ with a vertex v is thesimplicial complex defined byΣ ∗ v = Σ ∪ { σ ∪ { v } : σ ∈ Σ } ∪ . Evidently, we have that | Σ | ∗ v = | Σ ∗ v | . Proposition 2.1.
Let S , . . . , S n +1 ⊂ Y with ∅ = S ֒ → S ֒ → · · · n ֒ → S n +1 , and g : | Σ | → S n be a continuous map from an n -dimensional simplicial complex Σ such that g ( | Σ k | ) ⊂ S k , for every k ≤ n . If v / ∈ Σ , then g can be extended to acontinuous map h : | Σ ∗ v | → S n +1 such that h ( | (Σ ∗ v ) k | ) ⊂ S k , for every k ≤ n +1 .Proof. By a finite induction, extend each restriction g k = g ↾ (cid:12)(cid:12) Σ k (cid:12)(cid:12) to a continuousmap h k : (cid:12)(cid:12) (Σ ∗ v ) k (cid:12)(cid:12) → S k , k ≤ n , such that h k ↾ (cid:12)(cid:12) (Σ ∗ v ) k − (cid:12)(cid:12) = h k − , k >
0. Briefly,define h : (Σ ∗ v ) → S by h ↾ Σ = g and h ( v ) ∈ S . Whenever u ∈ Σ isa vertex of Σ, the map h : { u, v } → S can be extended to a continuous map h (1 ,u ) : |{ u, v }| → S because S ֒ → S . Then the map h : | (Σ ∗ v ) | → S definedby h ↾ (Σ ∗ v ) = h , h ↾ | Σ | = g and h ↾ |{ u, v }| = h (1 ,u ) , u ∈ Σ , is a continuousextension of both h and g . The construction can be carried on by induction to geta continuous extension h n : | (Σ ∗ v ) n | → S n of g = g n with the required properties.Finally, if σ ∈ Σ ∗ v is an ( n + 1)-dimensional simplex, then h n is defined on theboundary | σ | ∩ | (Σ ∗ v ) n | of | σ | which is homeomorphic to the n -sphere. Hence,it can be extended to a continuous map h σ : | σ | → S n +1 because S n n ֒ → S n +1 .The required map h : | Σ ∗ v | → S n +1 is now defined by h ↾ | (Σ ∗ v ) n | = h n and h ↾ | σ | = h σ , for every ( n + 1)-dimensional simplex σ ∈ Σ ∗ v . (cid:3)
3. Nerves of sequences of covers
The set Σ S of all nonempty finite subsets of a set S is a simplicial complex.Another natural example is the nerve of an indexed cover { U α : α ∈ A } of a set X , which is the subcomplex of Σ A defined by(3.1) N ( A ) = ( σ ∈ Σ A : \ α ∈ σ U α = ∅ ) . Following Lefschetz [12], the intersection T α ∈ σ U α is called the kernel of σ , andis often denoted by ker[ σ ] = T α ∈ σ U α . In case U is an unindexed cover of X , its ONSTRUCTING SELECTIONS STEPWISE OVER CONES 5 nerve is denoted by N ( U ). In this case, U is indexed by itself, and each simplex σ ∈ N ( U ) is merely a nonempty finite subset of U with ker[ σ ] = T σ = ∅ .Here, an important role will be played by a subcomplex of the nerve of a specialindexed cover of X . The prototype of this subcomplex can be found in some ofthe considerations in the proof of [15, Theorem 2.1]. Example 3.1.
Whenever 0 < κ ≤ ω , let U n , n < κ , be families of subsets of X such that S n<κ U n is a cover of X , and let F n<κ U n be the disjoint union of thesefamilies (obtained, for instance, by identifying each U n with U n × { n } , n < κ ).The nerve of this indexed cover of X defines a natural simplicial complex(3.2) N ( U <κ ) = N G n<κ U n ! . A simplex σ ∈ N ( U <κ ) can be described as the disjoint union σ = F mi =1 σ i of finitely many simplices σ i ∈ N ( U n i ), for n < · · · < n m < κ , such that T mi =1 ker[ σ i ] = ∅ . The simplicial complex N ( U <κ ) contains a natural subcomplex∆( U <κ ), define by(3.3) ∆( U <κ ) = (cid:8) σ ∈ N ( U <κ ) : card( σ ∩ U n × { n } ) ≤ , n < κ (cid:9) . In other words, the subcomplex ∆( U <κ ) consists of those simplices σ ∈ N ( U <κ )which are composed of finitely many vertices U i = ( U i , n i ) ∈ U n i , i ≤ m , where n < · · · < n m < κ . In the special case of κ = n + 1 < ω , we will simply write N ( U ≤ n ) = N ( U <κ ) and ∆( U ≤ n ) = ∆( U <κ ). (cid:3) The subcomplex ∆( U <κ ) in Example 3.1 is naturally related to the definitionof C -spaces. The following proposition is an immediate consequence of (3.3). Proposition 3.2.
Let < κ ≤ ω and V n , n < κ , be a sequence of pairwise-disjointfamilies of subsets of X , whose union forms a cover of X . Then (3.4) ∆( V <κ ) = N ( V <κ ) . For a simplicial complex Σ, a mapping Ω : X Σ will be called simplicial-valued if Ω( p ) is a subcomplex of Σ, for each p ∈ X . Such a mapping Ω : X Σgenerates a mapping | Ω | : X | Σ | defined by(3.5) | Ω | ( p ) = | Ω( p ) | = [ σ ∈ Ω( p ) | σ | , p ∈ X. Here is a natural example. Each indexed cover { U α : α ∈ A } of X generates anatural simplicial-valued mapping Σ A : X Σ A , defined by(3.6) Σ A ( p ) = ( σ ∈ Σ A : p ∈ \ α ∈ σ U α ) , p ∈ X. VALENTIN GUTEV
In fact, each Σ A ( p ) is a subcomplex of N ( A ), so Σ A : X N ( A ).The benefit of the mapping in (3.6) comes in the setting of the simplicial com-plex ∆( U <κ ) in Example 3.1 associated to a sequence of covers U n , n < κ , of X for some 0 < κ ≤ ω . Namely, we may define the corresponding simplicial-valuedmapping ∆ [ U <κ ] : X ∆( U <κ ) by the same pattern as in (3.6), i.e.(3.7) ∆ [ U <κ ] ( p ) = (cid:8) σ ∈ ∆( U <κ ) : p ∈ ker[ σ ] (cid:9) , p ∈ X. Just like before, we will write ∆ [ U ≤ n ] = ∆ [ U <κ ] whenever κ = n + 1 < ω .We now have the following natural relationship with aspherical sequences oflower locally constant mappings. Proposition 3.3.
Let U n , n < ω , be a sequence of covers of a set X . Then (3.8) ∆ [ U ≤ n ] ( p ) ∗ U ⊂ ∆ [ U ≤ n +1 ] ( p ) , whenever p ∈ U ∈ U n +1 and n < ω .Accordingly, (cid:12)(cid:12) ∆ [ U ≤ n ] (cid:12)(cid:12) : X | ∆( U ≤ n ) | ⊂ | ∆( U <ω ) | , n < ω , is an aspherical se-quence of mappings.Proof. The property in (3.8) follows from the fact that U = ( U, n + 1) / ∈ F nk =0 U k ,whenever p ∈ U ∈ U n +1 . Since (cid:12)(cid:12) ∆ [ U ≤ n ] ( p ) ∗ U (cid:12)(cid:12) = (cid:12)(cid:12) ∆ [ U ≤ n ] ( p ) (cid:12)(cid:12) ∗ U is contractible,this implies that (cid:12)(cid:12) ∆ [ U ≤ n ] ( p ) (cid:12)(cid:12) n ֒ → (cid:12)(cid:12) ∆ [ U ≤ n ] ( p ) ∗ U (cid:12)(cid:12) ⊂ (cid:12)(cid:12) ∆ [ U ≤ n +1 ] ( p ) (cid:12)(cid:12) . (cid:3) Proposition 3.4.
Let U , . . . , U n be a sequence of point-finite open covers of aspace X . Then the mapping (cid:12)(cid:12) ∆ [ U ≤ n ] (cid:12)(cid:12) : X | ∆( U ≤ n ) | is lower locally constant.Proof. Whenever p ∈ X , the set V p = T (cid:8) U ∈ S k ≤ n U k : p ∈ U (cid:9) is a neighbour-hood of p . Take a point q ∈ V p . Then by (3.7), σ ∈ ∆ [ U ≤ n ] ( p ) implies that σ ∈ ∆ [ U ≤ n ] ( q ) because q ∈ V p ⊂ ker[ σ ]. Thus, ∆ [ U ≤ n ] ( p ) ⊂ ∆ [ U ≤ n ] ( q ) and, accord-ingly, (cid:12)(cid:12) ∆ [ U ≤ n ] (cid:12)(cid:12) is lower locally constant. (cid:3) We conclude this section with a remark about the importance of disjoint unionsin the definition of the subcomplex ∆( U <κ ) in Example 3.1. Remark 3.5.
For a sequence of covers U n , n < κ , of X , where 0 < κ ≤ ω , onecan define the subcomplex ∆( U <κ ) ⊂ N ( U <κ ) by considering N ( U <κ ) to bethe nerve of the usual unindexed cover S n<κ U n , rather than the disjoint union F n<κ U n . However, this will not work to establish a property similar to that inProposition 3.3, also for the essential results in the next sections (see, for instance,Theorem 4.1 and Lemma 4.2). Namely, suppose that U and U are covers of X which contain elements U i ∈ U i , i = 0 ,
1, with U ∩ U = ∅ and U i / ∈ U − i .Then σ = { U , U } ∈ ∆( U ≤ ). However, if U is a cover of X with U , U ∈ U ,and ∆( U ≤ ) is defined on the basis of unindexed covers, then ∆( U ≤ ) ∆( U ≤ )because σ = { U , U } / ∈ ∆( U ≤ ). (cid:3) ONSTRUCTING SELECTIONS STEPWISE OVER CONES 7
4. Skeletal selections
For mappings ϕ, ψ : X Y , we will write ϕ ⊂ ψ to express that ϕ ( p ) ⊂ ψ ( p ),for every p ∈ X . In this case, the mapping ϕ is called a set-valued selection , ora multi-selection , for ψ . Also, let us recall that a cover F of a space X is called interior if the collection of the interiors of the elements of F is a cover of X .The following theorem will be proved in this section. Theorem 4.1.
Let X be a paracompact space and ϕ n : X Y , n < ω , be anaspherical sequence of lower locally constant mappings in a space Y . Then thereexists a sequence F n , n < ω , of closed locally finite interior covers of X and acontinuous map f : (cid:12)(cid:12) ∆( F <ω ) (cid:12)(cid:12) → Y such that (4.1) f ◦ (cid:12)(cid:12) ∆ [ F ≤ n ] (cid:12)(cid:12) ⊂ ϕ n , for every n < ω . Let us explicitly remark that, here, ∆ [ F ≤ n ] : X ∆( F ≤ n ) ⊂ ∆( F <ω ) is thesimplicial-valued mapping associated to the covers F k , k ≤ n , see (3.7), while f ◦ (cid:12)(cid:12) ∆ [ F ≤ n ] (cid:12)(cid:12) is the composite mapping | ∆( F <ω ) | X Y f (cid:12)(cid:12)(cid:12) ∆ [ F ≤ n ] (cid:12)(cid:12)(cid:12) f ◦ (cid:12)(cid:12)(cid:12) ∆ [ F ≤ n ] (cid:12)(cid:12)(cid:12) According to the definition of ∆ [ F ≤ n ] : X ∆( F ≤ n ), see also (3.5), the propertyin (4.1) means that f ( | σ | ) ⊂ ϕ n ( p ), for every σ ∈ ∆( F ≤ n ) and p ∈ ker[ σ ].Turning to the proof of Theorem 4.1, let us observe that the simplicial complex∆( F ≤ n ) is n -dimensional, see (3.3) of Example 3.1. In what follows, its k -skeletonwill be denoted by ∆ k ( F ≤ n ). In these terms, following the idea of an n -skeletalselection in [9], we shall say that a continuous map f : | ∆( F ≤ n ) | → Y is a skeletalselection for a sequence of mappings ϕ , . . . , ϕ n : X Y if(4.2) f ( | σ | ) ⊂ ϕ k ( p ) , for every σ ∈ ∆ k ( F ≤ n ), k ≤ n , and p ∈ ker[ σ ].Precisely as in (3.7), for each k ≤ n we may associate the simplicial-valued map-ping ∆ k [ F ≤ n ] : X ∆ k ( F ≤ n ), which assigns to each p in X the k -skeleton ∆ k [ F ≤ n ] ( p )of the subcomplex ∆ [ F ≤ n ] ( p ) ⊂ ∆( F ≤ n ). Then the property in (4.2) means thatthe composite mapping f ◦ (cid:12)(cid:12)(cid:12) ∆ k [ F ≤ n ] (cid:12)(cid:12)(cid:12) : X (cid:12)(cid:12) ∆ k ( F ≤ n ) (cid:12)(cid:12) is a set-valued selection for ϕ k , for every k ≤ n .Finally, let us recall that a simplicial map g : Σ → Σ is a map g : Σ → Σ between the vertices of simplicial complexes Σ and Σ such that g ( σ ) ∈ Σ , foreach σ ∈ Σ . If such a map g : Σ → Σ is bijective, then the inverse g − is also asimplicial map, and we say that g is a simplicial isomorphism . If g is only injective,then g embeds Σ into Σ , so that we may consider Σ as a subcomplex of Σ . VALENTIN GUTEV
Each simplicial map g : Σ → Σ generates a continuous map | g | : | Σ | → | Σ | which is affine on each geometric simplex | σ | , for σ ∈ Σ . Lemma 4.2.
Let Y be a space, F , . . . , F n be a sequence of closed locally finitecovers of a paracompact space X , and ϕ , . . . , ϕ n +1 : X Y be a sequence of lowerlocally constant mappings with ϕ k k ֒ → ϕ k +1 for every k ≤ n . If f n : | ∆( F ≤ n ) | → Y is a skeletal selection for ϕ , . . . , ϕ n , then there exists a closed locally finite interiorcover F n +1 of X and a continuous extension f n +1 : | ∆( F ≤ n +1 ) | → Y of f n whichis a skeletal selection for ϕ , . . . , ϕ n +1 .Proof. Let ∆ [ F ≤ n ] : X ∆( F ≤ n ) be the associated simplicial-valued mapping,defined as in (3.7). Whenever p ∈ X , the subcomplex(4.3) ∆ p = ∆ [ F ≤ n ] ( p )is n -dimensional such that, by (4.2), f n (cid:0)(cid:12)(cid:12) ∆ kp (cid:12)(cid:12)(cid:1) ⊂ ϕ k ( p ), k ≤ n . Moreover, by hy-pothesis, ϕ k ( p ) k ֒ → ϕ k +1 ( p ) for every k ≤ n . Since p / ∈ ∆ p , it follows from Propo-sition 2.1 that f n ↾ | ∆ p | can be extended to a continuous map f p : | ∆ p ∗ p | → Y such that(4.4) f p (cid:0)(cid:12)(cid:12) (∆ p ∗ p ) k (cid:12)(cid:12)(cid:1) ⊂ ϕ k ( p ) , for every 0 ≤ k ≤ n + 1.Since all covers are locally finite and closed, the point p ∈ X is contained in theopen set(4.5) O p = X \ [ { F ∈ F ∪ · · · ∪ F n : p / ∈ F } . For the same reason, ∆ p ∗ p is a finite simplicial complex. Accordingly, each set f p (cid:0)(cid:12)(cid:12) (∆ p ∗ p ) k (cid:12)(cid:12)(cid:1) , k ≤ n + 1, is compact. Hence, by (4.4) and the hypothesis thateach mapping ϕ k , k ≤ n + 1, is lower locally constant, we may shrink O p to aneighbourhood V p of p , defined by(4.6) V p = (cid:8) x ∈ O p : f p (cid:0)(cid:12)(cid:12) (∆ p ∗ p ) k (cid:12)(cid:12)(cid:1) ⊂ ϕ k ( x ) , for every k ≤ n + 1 (cid:9) . Finally, since X is paracompact, it has an open locally finite cover U n +1 such that { V p : p ∈ X } is refined by the associated cover F n +1 = (cid:8) U : U ∈ U n +1 (cid:9) of theclosures of the elements of U n +1 . So, there is a map p : F n +1 → X such that(4.7) F ⊂ V p ( F ) ⊂ O p ( F ) , for every F ∈ F n +1 .Having already defined the cover F n +1 , we are going to extend f n to a skeletalselection f n +1 : (cid:12)(cid:12) ∆( F ≤ n +1 ) (cid:12)(cid:12) → Y for the sequence ϕ , . . . , ϕ n +1 . To this end, takean F ∈ F n +1 , and define the set∆ F = (cid:8) τ ∈ ∆( F ≤ n ) : τ ∪ { F } ∈ ∆( F ≤ n +1 ) (cid:9) . It is evident that ∆ F is a subcomplex of ∆( F ≤ n ) with F / ∈ ∆ F , hence the cone∆ F ∗ F is a subcomplex of ∆( F ≤ n +1 ). Thus, to extend f n to a skeletal selection f n +1 : (cid:12)(cid:12) ∆( F ≤ n +1 ) (cid:12)(cid:12) → Y for the sequence ϕ , . . . , ϕ n +1 , it now suffices to extend ONSTRUCTING SELECTIONS STEPWISE OVER CONES 9 each f n ↾ | ∆ F | , F ∈ F n +1 , to a continuous map f F : | ∆ F ∗ F | → Y satisfying thecondition in (4.2) with respect to the simplices of ∆ F ∗ F . To this end, let usobserve that(4.8) ∆ F ⊂ ∆ p ( F ) = ∆ [ F ≤ n ] ( p ( F )) . Indeed, for T ∈ τ ∈ ∆ F , we have that ∅ = T ∩ F ⊂ T ∩ O p ( F ) , see (4.7). Hence,by (4.5), p ( F ) ∈ T and according to (3.7) and (4.3), τ ∈ ∆ p ( F ) .We are now ready to define the required maps f F : | ∆ F ∗ F | → Y , F ∈ F n +1 .Namely, by (4.8), we can embed ∆ F ∗ F into the cone ∆ p ( F ) ∗ p ( F ) by identifying p ( F ) with F . Let ℓ : ∆ F ∗ F → ∆ p ( F ) ∗ p ( F ) be the corresponding simplicialembedding defined by ℓ ↾ ∆ F to be the identity of ∆ F , and ℓ ( F ) = p ( F ). Next,define a continuous extension f F : | ∆ F ∗ F | → Y of f n ↾ | ∆ F | by f F = f p ( F ) ◦ | ℓ | .Take a simplex σ ∈ (∆ F ∗ F ) k for some k ≤ n +1, and a point x ∈ ker[ σ ]. If σ ∈ ∆ F ,by the properties of f n , see (4.2), f F ( | σ | ) = f n ( | σ | ) ⊂ ϕ k ( x ). If F ∈ σ , then x ∈ F ⊂ V p ( F ) and, by (4.6), we have again that f F ( | σ | ) = f p ( F ) ( | ℓ | ( | σ | )) ⊂ ϕ k ( x ).The proof is complete. (cid:3) Complementary to Lemma 4.2 is the following well-known property, see theproof of [15, Theorem 2.1] and that of [8, Theorem 3.1]. The property itself wasstated explicitly in [9, Proposition 3.2], and is an immediate consequence of thedefinition of lower locally constant mappings.
Proposition 4.3. If X is a paracompact space and ϕ : X Y is a lower locallyconstant mapping, then there exists a closed locally finite interior cover F of X and a ( continuous ) map f : ∆( F ) = F → Y such that f ( F ) ∈ ϕ ( x ) , for every x ∈ F ∈ F .Proof of Theorem 4.1. Inductively, using Proposition 4.3 and Lemma 4.2, thereexists a sequence F n , n < ω , of closed locally finite interior covers of X andcontinuous maps f n : (cid:12)(cid:12) ∆( F ≤ n ) (cid:12)(cid:12) → Y , n < ω , such that each f n is a skele-tal selection for the sequence ϕ , . . . , ϕ n , and each f n +1 is an extension of f n .Since ∆( F <ω ) = S n<ω ∆( F ≤ n ), we may define a map f : (cid:12)(cid:12) ∆( F <ω ) (cid:12)(cid:12) → Y by f ↾ (cid:12)(cid:12) ∆( F ≤ n ) (cid:12)(cid:12) = f n , for every n < ω . Then f is continuous and clearly has theproperty in (4.1). (cid:3)
5. Selections and canonical maps
Suppose that X is a (paracompact) space with the property that for any space Y , each aspherical sequence ϕ n : X Y , n < ω , of lower locally constant map-pings admits a continuous selection for its union S n<ω ϕ n . As we will see inthe next section (Corollaries 6.2, 6.3 and 6.5 and Example 6.4), each one of thefollowing statements determines a different dimension-like property of X . (5.1) There exists an aspherical sequence ϕ n : X Y , n < ω , of lower locallyconstant mappings such that no ϕ n , n < ω , has a continuous selection.(5.2) For each aspherical sequence ϕ k : X Y , k < ω , of lower locally constantmappings there exists an n < ω such that ϕ n has a continuous selection.(5.3) There exists an n < ω such that for each aspherical sequence ϕ k : X Y , k < ω , of lower locally constant mappings, the mapping ϕ n has a contin-uous selection.Here, we deal with the following general result reducing these selection problemsonly to simplicial-valued mappings associated to open locally finite covers of X . Theorem 5.1.
For a space Y , a paracompact space X and < µ ≤ ω + 1 , thefollowing are equivalent :(a) If ϕ n : X Y , n < ω , is an aspherical sequence of lower locally constantmappings, then S n<κ ϕ n has a continuous selection for some < κ < µ . (b) If U n , n < ω , is a sequence of open locally finite covers of X , then (cid:12)(cid:12) ∆ [ U <κ ] (cid:12)(cid:12) : X (cid:12)(cid:12) ∆( U <κ ) (cid:12)(cid:12) has a continuous selection for some < κ < µ . The proof of Theorem 5.1 is based on the results of the previous two sectionsand the following observation.
Proposition 5.2.
Let U n , n < ω , be a sequence of covers of X , < κ ≤ ω , and V n , n < κ , be a sequence of families of subsets of X such that each V n refines U n and S n<κ V n is a cover of X . If (cid:12)(cid:12) ∆ [ V <κ ] (cid:12)(cid:12) : X | ∆( V <κ ) | has a continuousselection, then so does (cid:12)(cid:12) ∆ [ U <κ ] (cid:12)(cid:12) : X | ∆( U <κ ) | .Proof. Since each V n refines U n , there are maps r n : V n → U n , n < κ , such that V ⊂ r n ( V ), for all V ∈ V n . Accordingly, r = F n<κ r n : ∆( V <κ ) → ∆( U <κ ) is asimplicial map with the property that σ ⊂ r ( σ ), for each simplex σ ∈ ∆( V <κ ). Inother words, r ◦ ∆ [ V <κ ] ⊂ ∆ [ U <κ ] , see (3.7), and therefore | r | ◦ (cid:12)(cid:12) ∆ [ V <κ ] (cid:12)(cid:12) ⊂ (cid:12)(cid:12) ∆ [ U <κ ] (cid:12)(cid:12) .Thus, if h : X → | ∆( V <κ ) | is a continuous selection for (cid:12)(cid:12) ∆ [ V <κ ] (cid:12)(cid:12) : X | ∆( V <κ ) | ,then the composite map f = | r | ◦ h : X → | ∆( U <κ ) | is a continuous selection for (cid:12)(cid:12) ∆ [ U <κ ] (cid:12)(cid:12) : X | ∆( U <κ ) | . (cid:3) Proof of Theorem 5.1.
The implication (a) = ⇒ (b) follows from Propositions 3.3and 3.4. The converse follows easily from Theorem 4.1 and Proposition 5.2.Namely, assume that (b) holds and ϕ n : X Y , n < ω , is as in (a). Since X is paracompact, by Theorem 4.1, there exists a sequence F n , n < ω , of closedlocally finite interior covers of X and a continuous map f : (cid:12)(cid:12) ∆ [ F <ω ] (cid:12)(cid:12) → Y satis-fying (4.1). For each n < ω , let U n be the cover of X composed by the interiorsof the elements of F n . Then by (b), the mapping (cid:12)(cid:12) ∆ [ U <κ ] (cid:12)(cid:12) : X (cid:12)(cid:12) ∆( U <κ ) (cid:12)(cid:12) hasa continuous selection for some 0 < κ < µ . According to Proposition 5.2, thisimplies that the mapping (cid:12)(cid:12) ∆ [ F <κ ] (cid:12)(cid:12) : X | ∆( F <κ ) | also has a continuous selection ONSTRUCTING SELECTIONS STEPWISE OVER CONES 11 h : X → (cid:12)(cid:12) ∆( F <κ ) (cid:12)(cid:12) . Evidently, the composite map g = f ◦ h : X → Y is acontinuous selection for the mapping S n<κ ϕ n . (cid:3) The selection problem in (b) of Theorem 5.1 is naturally related to the existenceof canonical maps for the disjoint union F n<κ U n of such covers. To this end, letus briefly recall some terminology. For a simplicial complex Σ and a simplex σ ∈ Σ, we use h σ i to denote the relative interior of the geometric simplex | σ | . Fora vertex v ∈ Σ , the set(5.4) st h v i = [ v ∈ σ ∈ Σ h σ i , is called the open star of the vertex v ∈ Σ . One can easily see that st h v i isopen in | Σ | because st h v i = | Σ | \ S v / ∈ σ ∈ Σ | σ | . In these terms, for an indexedcover { U α : α ∈ A } of a space X , a continuous map f : X → | N ( A ) | is called canonical for { U α : α ∈ A } if(5.5) f − (st h α i ) ⊂ U α , for every α ∈ A .It is well known that each open cover of a paracompact space admits a canonicalmap, which follows from the fact that such a cover has an index-subordinatedpartition of unity. The interested reader is referred to [9, Section 2] which containsa brief review of several facts about canonical maps and partitions of unity. Here,we are interested in a selection interpretation of canonical maps. Namely, interms of the simplicial-valued mapping Σ A : X N ( A ) associated to the cover { U α : α ∈ A } , see (3.6), we have the following characterisation of canonical maps;for unindexed covers it was obtained in [9, Proposition 2.5] (see also Dowker [4]),but the proof for indexed covers is essentially the same. Proposition 5.3.
A map f : X → | N ( A ) | is canonical for a cover { U α : α ∈ A } of a space X if and only if it is a continuous selection for the associated mapping | Σ A | : X | N ( A ) | . In the special case of a sequence of open covers U n , n < ω , a canonical map f : X → N ( U <ω ) for the disjoint union F n<ω U n will be called canonical for thesequence U n , n < ω . We now have the following further reduction of the selectionproblem for aspherical sequences of mappings, which is an immediate consequenceof Theorem 5.1 and Proposition 5.3. Corollary 5.4.
For a space Y , a paracompact space X and < µ ≤ ω + 1 , thefollowing are equivalent :(a) If ϕ n : X Y , n < ω , is an aspherical sequence of lower locally constantmappings, then S n<κ ϕ n has a continuous selection for some < κ < µ . (b) Each sequence U n , n < ω , of open covers of X admits a canonical map f : X → (cid:12)(cid:12) ∆( U <κ ) (cid:12)(cid:12) ⊂ | N ( U <ω ) | for some < κ < µ .
6. Dimension and canonical maps
Here, we finalise the proof of Theorem 1.2 by showing that the property C is equivalent to the existence of canonical maps for special covers. To this end,for a sequence U n , n < ω , of open covers of X and 0 < κ ≤ ω , we shall saythat a sequence V n , n < κ , of pairwise-disjoint families of open subsets X is a C -refinement of U n , n < ω , if each family V n refines U n and S n<κ V n covers X . Theorem 6.1.
For a paracompact space X and < µ ≤ ω + 1 , the following areequivalent :(a) Each sequence U n , n < ω , of open covers of X has a C -refinement V n , n < κ , for some < κ < µ . (b) Each sequence U n , n < ω , of open covers of X admits a canonical map f : X → (cid:12)(cid:12) ∆( U <κ ) (cid:12)(cid:12) for some < κ < µ .Proof. To see that (a) = ⇒ (b), take a sequence U n , n < ω , of open covers of X .Then by (a), U n , n < ω , admits a C -refinement V n , n < κ , for some 0 < κ < µ . Let N ( V <κ ) be the nerve of the disjoint union F n<κ V n , see (3.2) of Example 3.1, andΣ [ V <κ ] : X N ( V <κ ) be the simplicial-valued mapping associated to this nerve,see (3.6). Since X is paracompact, the indexed cover F n<κ V n has a canonical map.Hence, by Proposition 5.3, the mapping (cid:12)(cid:12) Σ [ V <κ ] (cid:12)(cid:12) : X | N ( V <κ ) | has a continu-ous selection. However, by definition, each family V n , n < κ , is pairwise-disjoint.Therefore, by (3.4) of Proposition 3.2, ∆( V <κ ) = N ( V <κ ) and, consequently,∆ [ V <κ ] = Σ [ V <κ ] . Thus, (cid:12)(cid:12) ∆ [ V <κ ] (cid:12)(cid:12) : X | ∆( V <κ ) | has a continuous selection and,according to Proposition 5.2, the mapping (cid:12)(cid:12) ∆ [ U <κ ] (cid:12)(cid:12) : X | ∆( U <κ ) | has a contin-uous selection as well. Finally, by Proposition 5.3, each continuous selection for (cid:12)(cid:12) ∆ [ U <κ ] (cid:12)(cid:12) is as required in (b).Conversely, let U n , n < ω , and f : X → | ∆( U <κ ) | be as in (b) for some0 < κ < µ . Define V n = { f − (st h U i ) : U ∈ U n } , n < κ . Since f is continuous, V n is an open family in X ; moreover, by (5.5), it refines U n . It is also evidentthat S n<µ V n covers X , see (5.4). We complete the proof by showing that V n ispairwise-disjoint as well. To this end, suppose that p ∈ f − (st h U i ) ∩ f − (st h U i )for some U , U ∈ U n and p ∈ X . Then f ( p ) ∈ st h U i ∩ st h U i and by (5.4), wehave that f ( p ) ∈ h σ i ∩ h σ i for some simplices σ , σ ∈ ∆( U <κ ) with U i ∈ σ i , i = 1 ,
2. Since the collection {h σ i : σ ∈ ∆( U <κ ) } forms a partition of | ∆( U <κ ) | ,this implies that σ = σ . Finally, according to the definition of ∆( U <µ ), see(3.3), we get that U = U . Thus, each family V n , n < µ , is also pairwise-disjoint,and the proof is complete. (cid:3) We finalise the paper with several applications. The first one is the followingslight generalisation of Theorem 1.2; it is an immediate consequence of Corol-lary 5.4 and Theorem 6.1 (in the special case of µ = ω + 1). ONSTRUCTING SELECTIONS STEPWISE OVER CONES 13
Corollary 6.2.
For a paracompact space X , the following are equivalent :(a) X is a C -space. (b) For every space Y , each aspherical sequence ϕ n : X Y , n < ω , oflower locally constant mappings admits a continuous selection for its union S n<ω ϕ n . (c) Each sequence U n , n < ω , of open covers of X admits a canonical map f : X → (cid:12)(cid:12) ∆( U <ω ) (cid:12)(cid:12) . Another consequence is for the case when 0 < µ = ω , and deals with the socalled finite C -spaces. These spaces were defined by Borst for separable metrizablespaces, see [2]; subsequently, the definition was extended by Valov [16] for arbitraryspaces. For simplicity, we will consider these spaces in the realm of normal spaces.In this setting, a (normal) space X is called a finite C -space if for any sequence { U k : k < ω } of finite open covers of X there exists a finite sequence { V k : k ≤ n } of open pairwise-disjoint families in X such that each V k refines U k and S k ≤ n V k is a cover of X . It was shown by Valov in [16, Theorem 2.4] that a paracompactspace X is a finite C -space if and only if each sequence { U k : k < ω } of open coversof X admits a finite C -refinement, i.e. there exists a finite sequence { V k : k ≤ n } of open pairwise-disjoint families in X such that each V k refines U k and S k ≤ n V k is a cover of X . Based on this, we have the following consequence of Corollary 5.4and Theorem 6.1 (in the special case of µ = ω ). Corollary 6.3.
For a paracompact space X , the following are equivalent :(a) X is a finite C -space. (b) For each aspherical sequence ϕ k : X Y , k < ω , of lower locally constantmappings in a space Y , there exists n < ω such that ϕ n has a continuousselection. (c) Each sequence U k , k < ω , of open covers of X admits a canonical map f : X → (cid:12)(cid:12) ∆( U ≤ n ) (cid:12)(cid:12) for some n < ω . Let us explicitly remark that the equivalence (a) ⇐⇒ (b) in Corollary 6.3 wasobtained by Valov in [16, Theorem 1.1]. His arguments were following those in [15]for proving Theorem 1.1. Accordingly, our approach is providing a simplificationof this proof. Regarding the proper place of finite C -spaces, it was shown byValov in [16, Proposition 2.2] that a Tychonoff space X is a finite C -space if andonly if its ˇCech-Stone compactification βX is a C -space. This brings a naturaldistinction between the selection problems stated in (5.1) and (5.2). Example 6.4.
The following example of a C -space which is not finite C was givenin [11, Remark 3.7]. Let K ω be the subspace of the Hilbert cube [0 , ω consistingof all points which have only finitely many nonzero coordinates. Then K ω is a C -space being strongly countable-dimensional, but is not a finite C -space becauseeach compactification of K ω contains a copy of [0 , ω (as per [5, Example 5.5.(1)]). According to Corollaries 6.2 and 6.3, see also Theorem 5.1, this implies that thereexists a space Y and an aspherical sequence ϕ n : K ω Y , n < ω , of lower locallyconstant mappings such that S n<ω ϕ n has a continuous selection, but none of themappings ϕ n , n < ω , has a continuous selection. (cid:3) Our last application is for the case when µ = n + 1 for some n < ω . To this end,following [9], a finite sequence ϕ k : X Y , 0 ≤ k ≤ n , of mappings will be called aspherical if ϕ k k ֒ → ϕ k +1 , for every k < n . By letting ϕ k ( p ) = Y ∗ q be the cone on Y with a fixed vertex q , where p ∈ X and k > n , each finite aspherical sequence ϕ k : X Y , 0 ≤ k ≤ n , can be extended to an aspherical sequence ϕ k : X Y ∗ q , k < ω . Furthermore, in this construction, each resulting new mapping ϕ k , k > n ,is lower locally constant being a constant set-valued mapping.Regarding dimension properties of the domain, let us recall a result of Ostrand[13] that for a normal space X with a covering dimension dim( X ) ≤ n , each openlocally finite cover U of X admits a sequence V , . . . , V n of open pairwise-disjointfamilies such that each V k refines U k and S nk =0 V k covers X . This result was refinedby Addis and Gresham, see [1, Proposition 2.12], that a paracompact space X hasa covering dimension dim( X ) ≤ n if and only if each finite sequence U , . . . , U n of open covers of X has a finite C -refinement, i.e. there exists a finite sequence V , . . . , V n of open pairwise-disjoint families of X such that each V k refines U k and S nk =0 V k covers X . Just like before, setting U k = U n , k > n , the abovecharacterisation of the covering dimension of paracompact spaces remains validfor an infinite sequence U k , k < ω , of open covers of X . Accordingly, we also havethe following consequence of Corollary 5.4 and Theorem 6.1 (in the special caseof µ = n + 1 < ω ). Corollary 6.5.
For a paracompact space X , the following are equivalent :(a) dim( X ) ≤ n . (b) For each aspherical sequence ϕ k : X Y , ≤ k ≤ n , of lower locally con-stant mappings in a space Y , the mapping ϕ n has a continuous selection. (c) Each sequence U k , ≤ k ≤ n , of open covers of X admits a canonicalmap f : X → (cid:12)(cid:12) ∆( U ≤ n ) (cid:12)(cid:12) . A direct poof of the implication (a) = ⇒ (b) in Corollary 6.5 was given in [9,Theorem 3.1]. Let us also remark that in the special case when all mappings ϕ k ,0 ≤ k ≤ n , are equal, the equivalence of (a) and (b) in Corollary 6.5 was shownin [15, Remark 2] and credited to Ernest Michael. References [1] D. Addis and J. Gresham,
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