Constructing Selections Stepwise Over Skeletons of Nerves of Covers
aa r X i v : . [ m a t h . GN ] O c t CONSTRUCTING SELECTIONS STEPWISE OVER SKELETONSOF NERVES OF COVERS
VALENTIN GUTEV
Abstract.
It is given a simplified and self-contained proof of the classicalMichael’s finite-dimensional selection theorem. The proof is based on approx-imate selections constructed stepwise over skeletons of nerves of covers. Themethod is also applied to simplify the proof of the Schepin–Brodsky’s general-isation of this theorem.
1. Introduction
All spaces in this paper are Hausdorff topological spaces. 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-valuedmapping . Such a mapping 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 . Also, let us recall that a map f : X → Y isa selection for Φ : X Y if f ( x ) ∈ Φ( x ), for all x ∈ X .Let n ≥ −
1. A family S of subsets of a space Y is equi- LC n [11] if everyneighbourhood U of a point y ∈ S S contains a neighbourhood V of y such thatfor every S ∈ S , every continuous map g : S k → V ∩ S of the k -sphere S k , k ≤ n ,can be extended to a continuous map h : B k +1 → U ∩ S of the ( k + 1)-ball B k +1 .A space S is called C n if for every k ≤ n , every continuous map g : S k → S can be extended to a continuous map h : B k +1 → S . In these terms, a family S of subsets of Y is equi- LC − if it consists of nonempty subsets; similarly, eachnonempty subset S ⊂ Y is C − .Let F ( Y ) be the collection of all nonempty closed subsets of a space Y . Thefollowing theorem was proved by Ernest Michael, see [11, Theorem 1.2], and iscommonly called the finite-dimensional selection theorem . Theorem 1.1.
Let X be a paracompact space with dim( X ) ≤ n + 1 , Y be a com-pletely metrizable space, and S ⊂ F ( Y ) be an equi- LC n family such that each S ∈ S is C n . Then each l.s.c. mapping Φ : X → S has a continuous selection. Date : October 16, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Lower semi-continuous mapping, lower locally constant mapping,continuous selection, local connectedness in finite dimension, finite-dimensional space.
The original proof of Theorem 1.1 in [11] takes up most of that paper, and isaccomplished in 6 steps. Other proofs of this theorem can be found in the mono-graph [12], and the book [7]. Actually, in [12] are given two different approachesto obtain the theorem — the one which follows the original Michael’s proof, andanother one based on filtrations [14]. Other proofs were given by other authors,see e.g. [1] and [9]. However, what all these proofs have in common is that theymay somehow discourage the casual reader and make Theorem 1.1 not so acces-sible to wider audience. The main purpose of this paper is to fill in this gap, andpresent a simplified and self-contained proof of this theorem.The paper is organised as follows. The next section contains a brief reviewof canonical maps and partitions of unity, which is essential for the proper un-derstanding of any of the available proofs of Theorem 1.1. In this regard, let usexplicitly remark that these considerations were not made readily available in pre-vious proofs, so they are now included to make the exposition self-contained. Theessential preparation for the proof of Theorem 1.1 starts in Section 3, which con-tains a selection theorem for finite aspherical sequences of lower locally constantmappings (Theorem 3.1). This theorem is similar to a theorem of Uspenskij, see[16, Theorem 1.3], and represents a relaxed version of another theorem proved bythe author, see [9, Theorem 3.1]. Section 4 contains several simple constructionsof finite aspherical sequences of sets providing the main interface between suchsequences of sets and the property of equi- LC n . Finally, the proof of Theorem1.1 is accomplished in Section 5. It is based on two constructions which are alsopresent in Michael’s proof. The one, Proposition 5.1, relates l.s.c. mappings tolower locally constant mappings; the other — Proposition 5.2, relates selectionsfor lower locally constant mappings to approximate selections for l.s.c. mappings.These constructions are applied together with Theorem 3.1 to deal with two se-lection properties of l.s.c. equi- LC n -valued mappings, see Theorems 5.3 and 5.4.The proof of Theorem 1.1 is then obtained as an immediate consequence of theseproperties.
2. Canonical maps and partitions of unity
The cozero set , or the set-theoretic support , of a function ξ : X → R is the setcoz( ξ ) = { x ∈ X : ξ ( x ) = 0 } . A collection ξ a : X → [0 , a ∈ A , of continuousfunctions on a space X is a partition of unity if P a ∈ A ξ a ( x ) = 1, for each x ∈ X .Here, “ P a ∈ A ξ a ( x ) = 1” means that only countably many functions ξ a ’s do notvanish at x , and the series composed by them is convergent to 1. For a cover U of a space X , a partition of unity { ξ U : U ∈ U } on X is index-subordinated to U if coz( ξ U ) ⊂ U , for each U ∈ U , see Remark 2.7. The following theorem is wellknown, it is a consequence of Urysohn’s characterisation of normality [15] and theLefschetz lemma [10]. ONSTRUCTING SELECTIONS STEPWISE OVER SKELETONS OF NERVES 3
Theorem 2.1.
Every locally finite open cover of a normal space has an index-subordinated partition of unity.
A partition of unity { ξ a : a ∈ A } on a space X is called locally finite if { coz( ξ a ) : a ∈ A } is a locally finite cover of X . Complementary to Theorem2.1 is the following important property of partitions of unity; it follows from aconstruction of M. Mather, see [3, Lemma] and [6, Lemma 5.1.8]. Theorem 2.2.
If a cover U of a space X has an index-subordinated partition ofunity, then U also has an index-subordinated locally finite partition of unity. 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 . In these terms, 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 , for each σ ∈ Σ . If g : Σ → Σ is a simplicial map and g : Σ → Σ is bijective, then the inverse g − is also a simplicial map, and we say that g is a simplicial isomorphism .The set Σ S of all nonempty finite subsets of a set S is a simplicial complex.Another natural example is the nerve N ( U ) of a cover U of a set X , which isthe subcomplex of Σ U defined by(2.1) N ( U ) = n σ ∈ Σ U : \ σ = ∅ o . The k -skeleton of N ( U ) is denoted by N k ( U ), and the vertex set N ( U ) of N ( U ) is actually U because we can always assume that ∅ / ∈ U .For a set A , let ℓ ( A ) be the linear space of all functions y : A → R with P a ∈ A | y ( a ) | < ∞ . In fact, ℓ ( A ) is a Banach space when equipped with the norm k y k = P a ∈ A | y ( a ) | , but this will play no role in the paper. The vertex set Σ of a simplicial complex Σ is a linearly independent subset of ℓ (Σ ), where each v ∈ Σ is identified with its characteristic function v : Σ → { , } , namely withthe function v ( u ) = 0 for u = v , and v ( v ) = 1. Then to each σ ∈ Σ one canassociate the geometric simplex | σ | = conv( σ ), which is the convex hull of σ .Thus, | σ | is a k -dimensional simplex if and only if Card( σ ) = k + 1. The set | Σ | = S σ ∈ Σ | σ | ⊂ ℓ (Σ ) is called the geometric realisation of Σ. As a topologicalspace, we will consider | Σ | endowed with the Whitehead topology [18, 19]. Inthis topology, a subset U ⊂ | Σ | is open if and only if U ∩ | σ | is open in | σ | , forevery σ ∈ Σ. Let us explicitly remark that the Whitehead topology on | Σ | is notnecessarily the subspace topology on | Σ | as a subset of the Banach space ℓ (Σ ).However, both topologies coincide on each geometric simplex | σ | , for σ ∈ Σ. VALENTIN GUTEV If p ∈ | σ | for some σ ∈ Σ, then p is both an element p ∈ ℓ (Σ ) and a uniqueconvex combination of the elements of σ ⊂ Σ ⊂ ℓ (Σ ). Hence, the geometricrealisation | Σ | is the set of all p ∈ ℓ (Σ ) such that(2.2) p ( v ) ≥ , v ∈ Σ , and coz( p ) = (cid:8) v ∈ Σ : p ( v ) > (cid:9) ∈ Σ . Here, p ( v ) is called the v -th barycentric (or affine ) coordinate of p ∈ | Σ | , whilethe simplex coz( p ) ∈ Σ is called the carrier of p , and denoted by car( p ) = coz( p ).Since the representation p = P v ∈ car( p ) p ( v ) · v is unique, the carrier car( p ) is theminimal simplex of Σ with the property that p ∈ | car( p ) | .To each vertex v ∈ Σ , we can now associate the function α v : | Σ | → [0 , α v ( p ) = p ( v ) , for every p ∈ | Σ | .It is called the v -th barycentric coordinate function and is continuous being affineon each simplex | σ | , for σ ∈ Σ. The cozero set coz( α v ) of α v is called the openstar of the vertex v ∈ Σ , and denoted by(2.4) st h v i = (cid:8) p ∈ | Σ | : α v ( p ) > (cid:9) . Clearly, the open star st h v i is open in | Σ | because α v is continuous. The followingproposition is an immediate consequence of (2.2), (2.3) and (2.4). Proposition 2.3. If Σ is a simplicial complex, then the collection { α v : v ∈ Σ } is a partition of unity on | Σ | with coz( α v ) = st h v i , for each v ∈ Σ . We now turn to the other essential concept in this section. For a cover U of aspace X , a continuous map f : X → | N ( U ) | is called canonical for U if(2.5) f − (st h U i ) ⊂ U, for every U ∈ U .Canonical maps are essentially partitions of unity, which are index-subordinatedto the corresponding cover of the space. Theorem 2.4.
A cover U of a space X has an index-subordinated partition ofunity if and only if U has a canonical map.Proof. Let U be a cover of X and α U , U ∈ U , be the barycentric coordinatefunctions of | N ( U ) | .Suppose that f : X → | N ( U ) | is a canonical map for U . Since f is continuous,by Proposition 2.3, { α U ◦ f : U ∈ U } is a partition of unity on X . By the sameproposition and (2.5), we also have thatcoz( α U ◦ f ) = f − (coz( α U )) = f − (st h U i ) ⊂ U, U ∈ U . Conversely, suppose that U has an index-subordinated partition of unity. Thenby Theorem 2.2, U also has an index-subordinated locally finite partition of unity { ξ U : U ∈ U } . For each x ∈ X , let σ ξ ( x ) ∈ N ( U ) be the simplex determined by ONSTRUCTING SELECTIONS STEPWISE OVER SKELETONS OF NERVES 5 the point x and the functions ξ U , U ∈ U , namely σ ξ ( x ) = { U ∈ U : ξ U ( x ) > } .Next, define a map f : X → | N ( U ) | by(2.6) f ( x ) = X U ∈ σ ξ ( x ) ξ U ( x ) · U, x ∈ X. Since { ξ U : U ∈ U } is a locally finite partition of unity, each point p ∈ X has aneighbourhood V p ⊂ X such that U p = { U ∈ U : V p ∩ coz( ξ U ) = ∅ } is a finite set.According to (2.6), this implies that f ( V p ) ⊂ | N ( U p ) | ⊂ ℓ ( U p ). However, ℓ ( U p )is now the usual Euclidean space R U p because U p is a finite set. For the samereason, N ( U p ) has finitely many simplices. Therefore, the Whitehead topologyon | N ( U p ) | is the subspace topology on | N ( U p ) | as a subset of R U p . Since eachfunction ξ U = α U ◦ f , U ∈ U p , is continuous, so is the restriction f ↾ V p . Thisshows that f is continuous as well. Finally, let U ∈ U and x ∈ f − (st h U i ).Then f ( x ) ∈ st h U i and by (2.4) and (2.6), we get that ξ U ( x ) = α U ( f ( x )) > U ∈ σ ξ ( x ) which implies that x ∈ U because coz( ξ U ) ⊂ U . Thus, f is canonical for U , see (2.5). (cid:3) Canonical maps will be involved in the proof of Theorem 1.1 with two properties,which are briefly discussed below.For a simplicial complex Σ, as mentioned before, the carrier car( p ) of a point p ∈ | Σ | is the minimal simplex of Σ with p ∈ | car( p ) | , see (2.2). According to(2.3) and (2.4), it has the following natural representation(2.7) car( p ) = (cid:8) v ∈ Σ : p ∈ st h v i (cid:9) . For a cover U of X and x ∈ X , we will associate the simplicial complex(2.8) Σ U ( x ) = n σ ∈ Σ U : x ∈ \ σ o . According to (2.1), we have that Σ U ( x ) ⊂ N ( U ), for every x ∈ X . Thus, (2.8)defines a natural set-valued mapping Σ U : X N ( U ). To this mapping, wewill associate the mapping | Σ U | : X | N ( U ) | which assigns to each x ∈ X thegeometric realisation | Σ U | ( x ) = | Σ U ( x ) | . In terms of this mapping, we have thefollowing selection interpretation of canonical maps which extends an observationof Dowker [4], see Remark 2.9. Proposition 2.5.
Let U be a cover of a space X . Then a continuous map f : X → | N ( U ) | is canonical for U if and only if f is a selection for themapping | Σ U | : X | N ( U ) | .Proof. Let f be a canonical map for U , and x ∈ X . Whenever U ∈ car( f ( x )), itfollows from (2.7) that f ( x ) ∈ st h U i and therefore, by (2.5), x ∈ U . Thus, by (2.8),car( f ( x )) ∈ Σ U ( x ) and we have that f ( x ) ∈ | car( f ( x )) | ⊂ | Σ U | ( x ). Conversely,suppose that f is as selection for | Σ U | , and x ∈ f − (st h U i ) for some U ∈ U . VALENTIN GUTEV
Then by (2.7), U ∈ car( f ( x )) because f ( x ) ∈ st h U i . Moreover, f ( x ) ∈ | σ | forsome σ ∈ Σ U ( x ) because f ( x ) ∈ | Σ U | ( x ). Since car( f ( x )) is the minimal simplexwith this property, we get that U ∈ car( f ( x )) ⊂ σ and, therefore, x ∈ U . That is, f − (st h U i ) ⊂ U . (cid:3) Each simplicial map g : Σ → Σ , between simplicial complexes Σ and Σ ,can be extended to a continuous map | g | : | Σ | → | Σ | which is affine on eachgeometric simplex | σ | , for σ ∈ Σ . This map is simply defined by | g | ( p ) = X v ∈ car( p ) α v ( p ) · g ( v ) , p ∈ | Σ | . If a cover V of X refines another cover U , then there exists a natural simplicialmap r : N ( V ) → N ( U ) with V ⊂ r ( V ), for each V ∈ V . Such a map iscommonly called a canonical projection , or a refining simplicial map , or simply a refining map . Canonical maps are preserved by refinements in the following sense. Corollary 2.6.
Let U and V be covers of a space X such that V refines U . If r : N ( V ) → N ( U ) is a refining map and g : X → | N ( V ) | is canonical for V ,then the composite map | r | ◦ g : X → | N ( U ) | is canonical for U .Proof. This follows from Proposition 2.5 and the fact that r (Σ V ( x )) ⊂ Σ U ( x ), x ∈ X , because V ⊂ r ( V ) for every V ∈ V , see (2.8). (cid:3) We conclude this section with several remarks.
Remark 2.7.
For a space X , the support of a function ξ : X → R , called alsothe topological support , is the set supp( ξ ) = coz( ξ ). In several sources, a partitionof unity { ξ U : U ∈ U } on a space X is called index-subordinated to a cover U of X if supp( ξ U ) ⊂ U , for every U ∈ U ; and { ξ U : U ∈ U } is called weakly index-subordinated to U if coz( ξ U ) ⊂ U , for every U ∈ U , see e.g. [13]. However, thesevariations in the terminology do not affect the results of this section. Namely,if { η U : U ∈ U } is a partition of unity on X , then X also has a (locally finite)partition of unity { ξ U : U ∈ U } with supp( ξ U ) ⊂ coz( η U ), for all U ∈ U , [13,Proposition 2.7.4]. This property is essentially the construction of M. Mather forproving Theorem 2.2. Remark 2.8.
Canonical maps provide an isomorphism between simplicial com-plexes and nerves of covers. Namely, if O Σ = (cid:8) st h v i : v ∈ Σ (cid:9) is the coverof | Σ | by the open stars of the vertices of a simplicial complex Σ and σ ⊂ Σ ,then σ ∈ Σ if and only if T v ∈ σ st h v i 6 = ∅ . That is, σ ∈ Σ precisely whenst h σ i = { st h v i : v ∈ σ } ∈ N ( O Σ ). Hence, st h·i : Σ → N ( O Σ ) is a simplicialisomorphism and the associated map | st h·i| : | Σ | → | N ( O Σ ) | is both a homeo-morphism and a canonical map for O Σ . ONSTRUCTING SELECTIONS STEPWISE OVER SKELETONS OF NERVES 7
Remark 2.9.
In the case of a point-finite cover U of X , Proposition 2.5 is reducedto the following selection interpretation of canonical maps given by Dowker [4].Whenever x ∈ X , let σ ( x ) = { U ∈ U : x ∈ U } ∈ N ( U ) be the simplexdetermined by x . Then a continuous map f : X → | N ( U ) | is canonical for U if and only if f ( x ) ∈ | σ ( x ) | , for every x ∈ X . While σ ( x ) is only an element ofΣ U ( x ), we have that | σ ( x ) | = | Σ U | ( x ) because σ ⊂ σ ( x ), for each σ ∈ Σ U ( x ).
3. Aspherical sequences of mappings and selections
A mapping ϕ : X Y is lower locally constant [9] if the set { x ∈ X : K ⊂ ϕ ( x ) } is open in X , for every compact subset K ⊂ Y . This property appeared in apaper of Uspenskij [16]; later on, it was used by some authors (see, for instance,[2, 17]) under the name “strongly l.s.c.”, while in papers of other authors stronglyl.s.c. was already used for a different property of set-valued mappings (see, forinstance, [8]). Every lower locally constant mapping is l.s.c. but the converse failsin general and counterexamples abound. In fact, if we consider a single-valuedmap 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 if and only if it is locally constant. Thus,our terminology provides some natural analogy with the single-valued case.Let k ≥
0. For subsets
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 . Evidently, the relation S k ֒ → B implies that S ⊂ 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 of mappings ϕ k : X Y , 0 ≤ k ≤ n ,is aspherical if ϕ k k ֒ → ϕ k +1 , for every k < n . The following theorem will be provedin this section. Theorem 3.1.
Let X be a paracompact space with dim( X ) ≤ n , Y be a space,and ϕ k : X Y , ≤ k ≤ n , be an aspherical sequence of lower locally constantmappings. Then ϕ n has a continuous selection. The proof of Theorem 3.1 is based on special skeletal selections motivated bythe characterisation of canonical maps in Proposition 2.5. Namely, we shall saythat a mapping ϕ : X Y has a k -skeletal selection , k ≥
0, if there exists andopen cover U of X and a continuous map u : | N k ( U ) | → Y such that(3.1) u (cid:0) | Σ k U ( x ) | (cid:1) ⊂ ϕ ( x ) , for every x ∈ X .Here, Σ k U ( x ) is the k -skeleton of the simplicial complex Σ U ( x ), see (2.8). In fact,just like before, one can consider Σ k U : X N k ( U ) as a set-valued mapping;similarly for | Σ k U | : X | N k ( U ) | . Then a continuous map u : | N k ( U ) | → Y isa k -skeletal selection for ϕ if and only if the composite mapping u ◦ | Σ k U | : X Y is a set-valued selection for ϕ : X Y , see Remark 3.6. VALENTIN GUTEV
We proceed with the following constructions of k -skeletal selections which fur-nish the essential part of the proof of Theorem 3.1. Proposition 3.2.
Each lower locally constant mapping ϕ : X Y has a -skeletalselection.Proof. For each x ∈ X , take a point y ( x ) ∈ ϕ ( x ), and set(3.2) U ( x ) = (cid:8) z ∈ X : y ( x ) ∈ ϕ ( z ) (cid:9) . Then U = { U ( x ) : x ∈ X } is an open cover of X . Moreover, for each U ∈ U there is a point x U ∈ X with U = U ( x U ). Since | N ( U ) | = U , we may define amap u : | N ( U ) | → Y by u ( U ) = y ( x U ), for each U ∈ U . If x ∈ U ∈ U , then x ∈ U ( x U ) and by (3.2), we get that u ( U ) = y ( x U ) ∈ ϕ ( x ). (cid:3) Proposition 3.3.
Let X be a paracompact space, and ψ : X Y be a mappingwhich has a k -skeletal selection, for some k ≥ . Then ψ has a k -skeletal selection u : | N k ( U ) | → Y for some open locally finite cover U of X .Proof. Let v : | N k ( V ) | → Y be a k -skeletal selection for ψ , for some opencover V of X . Since X is paracompact, the cover V has an open locally finiterefinement U . Let r : N ( U ) → N ( V ) be a refining map. Then by (3.1), u = v ◦ | r | ↾ | N k ( U ) | : | N k ( U ) | → Y is a k -skeletal selection for ψ because r (Σ k U ( x )) ⊂ Σ k V ( x ), for every x ∈ X . (cid:3) A cover V of X is a star-refinement of a cover U if the cover V ∗ = { V ∗ : V ∈ V } refines U , where V ∗ = S { W ∈ V : W ∩ V = ∅ } . To reflect this property, weshall say that a simplicial map ℓ : N ( V ) → N ( U ) is a star-refining map if V ∗ ⊂ ℓ ( V ), for each V ∈ V . Each star-refining map ℓ : N ( V ) → N ( U ) has theproperty that(3.3) [ σ ⊂ \ ℓ ( σ ) , for each σ ∈ N ( V ). Proposition 3.4.
Let X be a paracompact space, Y be a space, and ψ, ϕ : X Y be such that ϕ is lower locally constant and ψ k ֒ → ϕ for some k ≥ . If ψ has a k -skeletal selection, then ϕ has a ( k + 1) -skeletal selection.Proof. By Proposition 3.3, ψ has a k -skeletal selection u : | N k ( U ) | → Y for someopen locally finite cover U of X . For each σ ∈ N ( U ), let u σ = u ↾ | N k ( σ ) | be therestriction of u over the subcomplex | N k ( σ ) | = | σ | ∩ | N k ( U ) | . Then, whenever σ ∈ Σ k +1 U ( x ) for some x ∈ X , the map u σ can be extended to a continuous map u ( x,σ ) : | σ | → Y such that(3.4) u ( x,σ ) ( | σ | ) ⊂ ϕ ( x ) . Indeed, if σ ∈ Σ k U ( x ), then by (3.1), u ( | σ | ) ⊂ ψ ( x ) ⊂ ϕ ( x ) and we can take u ( x,σ ) = u σ . If σ / ∈ Σ k U ( x ), then | N k ( σ ) | = S {| τ | : ∅ = τ ( σ } is homeomorphic ONSTRUCTING SELECTIONS STEPWISE OVER SKELETONS OF NERVES 9 to the k -sphere being the boundary of | σ | . Hence, u σ has a continuous extension u ( x,σ ) : | σ | → ϕ ( x ) because u ( | N k ( σ ) | ) ⊂ u ( | Σ k U ( x ) | ) ⊂ ψ ( x ) k ֒ → ϕ ( x ), see (3.1).Now, whenever x ∈ X , set(3.5) K ( x ) = [ (cid:8) u ( x,σ ) ( | σ | ) : σ ∈ Σ k +1 U ( x ) (cid:9) . Then by (3.4), K ( x ) ⊂ ϕ ( x ); moreover, K ( x ) is compact because U is locallyfinite and, therefore, Σ k +1 U ( x ) contains finitely many simplices. Since ϕ is lowerlocally constant, for each U ∈ U , each point x ∈ U is contained in the open set(3.6) W ( x,U ) = (cid:8) z ∈ U : K ( x ) ⊂ ϕ ( z ) (cid:9) . Since X is paracompact, the cover { W ( x,U ) : x ∈ U ∈ U } has an open star-refinement V . So, there are maps p : V → X and ℓ : V → U such that(3.7) V ∗ ⊂ W ( p ( V ) ,ℓ ( V )) , for every V ∈ V .Accordingly, ℓ is a star-refining map because by (3.6), V ∗ ⊂ W ( p ( V ) ,ℓ ( V )) ⊂ ℓ ( V ).Finally, take a map q : N ( V ) → V which selects from any simplex σ ∈ N ( V ) avertex q ( σ ) ∈ σ , and next set π = p ◦ q : N ( V ) → X . Then(3.8) ℓ ( σ ) ∈ Σ U ( π ( σ )) , σ ∈ N ( V ) , because π ( σ ) = p ( q ( σ )) ∈ q ( σ ) ⊂ S σ ⊂ T ℓ ( σ ), see (3.3).We complete the proof as follows. Using (3.4) and (3.8), one can define a contin-uous extension v : | N k +1 ( V ) | → Y of the map u ◦ | ℓ | ↾ | N k ( V ) | : | N k ( V ) | → Y by v ↾ | σ | = u ( π ( σ ) ,ℓ ( σ )) , for every σ ∈ N k +1 ( V ). This v is a ( k + 1)-skeletal se-lection for ϕ . Indeed, let σ ∈ Σ k +1 V ( x ) for some x ∈ X . Then x ∈ q ( σ ) because q ( σ ) ∈ σ , see (2.8). Moreover, by (3.7), q ( σ ) ⊂ [ q ( σ )] ∗ ⊂ W ( π ( σ ) ,ℓ ( q ( σ ))) . Hence, by(3.5), (3.6) and (3.8), v ( | σ | ) = u ( π ( σ ) ,ℓ ( σ )) ( | ℓ ( σ ) | ) ⊂ K ( π ( σ )) ⊂ ϕ ( x ). (cid:3) Proof of Theorem 3.1.
According to Propositions 3.2 and 3.4, the mapping ϕ n hasan n -skeletal selection u : | N n ( U ) | → Y , for some open cover U of X . Since X is paracompact and dim( X ) ≤ n , the cover U has an open refinement V with N ( V ) = N n ( V ), see Remark 3.5. Let r : N ( V ) → N n ( U ) be a refiningmap, and g : X → | N ( V ) | be a canonical map for V which exists because X isparacompact, see Theorems 2.1 and 2.4. Then by Corollary 2.6, the compositemap h = | r | ◦ g : X → | N n ( U ) | is a canonical map for U . Finally, by (3.1) andProposition 2.5, the composite map f = u ◦ h : X → Y | N n ( U ) | X Y uh ϕ is a continuous selection for ϕ . (cid:3) Remark 3.5.
Let n ≥ − X be a normal space. The order of acover V of X doesn’t exceed n if T σ = ∅ , for every σ ⊂ V with Card( σ ) ≥ n + 2;equivalently, if N ( V ) = N n ( V ). In these terms, the covering dimension of X is at most n , written dim( X ) ≤ n , if every finite open cover of X has anopen refinement V with N ( V ) = N n ( V ). According to a result of Dowker [4,Theorem 3.5], dim( X ) ≤ n if and only if every locally finite open cover of X hasan open refinement V with N ( V ) = N n ( V ). In particular, for a paracompactspace X , we have that dim( X ) ≤ n if and only if every open cover of X has anopen refinement V with N ( V ) = N n ( V ). Remark 3.6.
A mapping ψ : X Y is a set-valued selection (or set-selection ,or multi-selection ) for ϕ : X Y if ψ ( x ) ⊂ ϕ ( x ), for all x ∈ X . In terms ofset-valued selections, a mapping ϕ : X Y has a k -skeletal selection, k ≥
0, ifthere exists an open cover U of X and a continuous map u : | N k ( U ) | → Y suchthat the composite mapping u ◦ | Σ k U | : X Y | N k ( U ) | X Y u | Σ k U | ϕ is a set-valued selection for ϕ : X Y .
4. Generating aspherical sequences of sets
For a point y ∈ Y of a metric space ( Y, d ) and ε >
0, let O ε ( y ) = { z ∈ Y : d ( z, y ) < ε } be the open ε -ball centred at y ; and O ε ( S ) = S y ∈ S O ε ( y ) be the ε -neighbourhoodof a subset S ⊂ Y . Also, recall that 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 .Throughout this section, δ : (0 , + ∞ ) → (0 , + ∞ ) is a fixed function. To thisfunction, we associate the sequence of iterated functions δ n : (0 , + ∞ ) → (0 , + ∞ ), n ≥
0, defined by(4.1) δ ( ε ) = ε and δ n +1 ( ε ) = δ ( δ n ( ε )) . Proposition 4.1.
Let ( Y, d ) be a metric space and S ⊂ S ⊂ · · · ⊂ S n ⊂ Y besuch that O δ ( ε ) ( y ) ∩ S k k ֒ → O ε ( y ) ∩ S k +1 , for every y ∈ Y and k < n . Then (4.2) O δ n − k ( ε ) ( y ) ∩ S k k ֒ → O δ n − k − ( ε ) ( y ) ∩ S k +1 , k < n. Proof.
Follows from the fact that δ n − k ( ε ) = δ (cid:0) δ n − k − ( ε ) (cid:1) , see (4.1). (cid:3) We now have the following “local” version of Theorem 3.1.
ONSTRUCTING SELECTIONS STEPWISE OVER SKELETONS OF NERVES 11
Theorem 4.2.
Let ( Y, d ) be a metric space, X be a paracompact space with dim( X ) ≤ n , and ψ k : X Y , ≤ k ≤ n , be lower locally constant mappingssuch that O δ ( ε ) ( y ) ∩ ψ k ( x ) k ֒ → O ε ( y ) ∩ ψ k +1 ( x ) for every x ∈ X , y ∈ Y and k < n .Then for each continuous δ n ( ε ) -selection g : X → Y for ψ , there is a continuousselection f : X → Y for ψ n with d ( f ( x ) , g ( x )) < ε , for all x ∈ X .Proof. Let g : X → Y be a continuous δ n ( ε )-selection for ψ . Next, for each k ≤ n ,define a set-valued mapping ϕ k by ϕ k ( x ) = O δ n − k ( ε ) ( g ( x )) ∩ ψ k ( x ), x ∈ X . Since g is a δ n ( ε )-selection for ψ , the mapping ϕ is nonempty-valued and, according to(4.2), so is each ϕ k , k ≤ n . In fact, by (4.2), the resulting sequence of mappings ϕ k : X Y , 0 ≤ k ≤ n , is aspherical. Moreover, each ϕ k is lower locally constantbecause so are ψ k and the mapping x → O δ n − k ( ε ) ( g ( x )), x ∈ X (see Proposition5.1). Hence, by Theorem 3.1, ϕ n has a continuous selection f : X → Y because X is a paracompact space with dim( X ) ≤ n . Evidently, f is a selection for ψ n and, by (4.1), f ( x ) ∈ O δ n − n ( ε ) ( g ( x )) = O δ ( ε ) ( g ( x )) = O ε ( g ( x )), x ∈ X . (cid:3) We conclude this section with the following two applications of Theorem 4.2which will provide the main interface between selections for l.s.c. mappings andTheorem 3.1, see Theorems 5.3 and 5.4.
Corollary 4.3.
Let E be a normed space, and ∅ = S ⊂ T ⊂ E be such that S k ֒ → T and O δ ( ε ) ( y ) ∩ S i ֒ → O ε ( y ) ∩ S , for every y ∈ E and ≤ i < k . Then O δ k ( ε ) ( S ) k ֒ → O ε ( T ) .Proof. Let ℓ : S k → O δ k ( ε ) ( S ) be a continuous map from the k -sphere S k . Considerthe constant mappings ψ i ( x ) = S , x ∈ S k and i ≤ k . Then ℓ is a continuous δ k ( ε )-selection for ψ , and O δ ( ε ) ( y ) ∩ ψ i ( x ) i ֒ → O ε ( y ) ∩ ψ i +1 ( x ) for every x ∈ S k , y ∈ E and i < k . Hence, by Theorem 4.2, there exists a continuous q : S k → S with k q ( x ) − ℓ ( x ) k < ε , for every x ∈ S k . Let h be the linear homotopy between ℓ and q , i.e. h ( x, t ) = tq ( x ) + (1 − t ) ℓ ( x ), whenever ( x, t ) ∈ S k × [0 , h ( S k × [0 , ⊂ O ε ( S ) ⊂ O ε ( T ). Also, let h : S k × [0 , → T be a homotopybetween q and a constant map, which exists because S k ֒ → T . Finally, take h tobe the homotopy obtained by combining h and h . Then h is a homotopy of ℓ with a constant map over a subset of O ε ( T ). (cid:3) Corollary 4.4.
Let E be a normed space, and ∅ = S ⊂ T ⊂ E be such that O δ ( ε ) ( y ) ∩ T k ֒ → O ε ( y ) ∩ T and O δ ( ε ) ( y ) ∩ S i ֒ → O ε ( y ) ∩ S , for every y ∈ E and ≤ i < k . Define functions (4.3) η ( ε ) = δ ( ε ) / and λ ( ε, µ ) = δ k (cid:0) min { η ( ε ) , µ } (cid:1) , ε, µ > . Then O η ( ε ) ( y ) ∩ O λ ( ε,µ ) ( S ) k ֒ → O ε ( y ) ∩ O µ ( T ) , for every y ∈ E . Proof.
Let ℓ : S k → O η ( ε ) ( y ) ∩ O λ ( ε,µ ) ( S ) be a continuous map for some y ∈ E .Then, precisely as in the previous proof, there exists a continuous map q : S k → S such that k q ( x ) − ℓ ( x ) k < min { η ( ε ) , µ } , for every x ∈ S k . Since η ( ε ) = δ ( ε ) /
2, see(4.3), just like before, using a linear homotopy, we get that ℓ and q are homotopicin O δ ( ε ) ( y ) ∩ O µ ( S ). Moreover q is homotopic to a constant map in O ε ( y ) ∩ T because q : S k → O δ ( ε ) ( y ) ∩ S ⊂ O δ ( ε ) ( y ) ∩ T k ֒ → O ε ( y ) ∩ T . Accordingly, ℓ ishomotopic to a constant map in O ε ( y ) ∩ O µ ( T ). (cid:3)
5. Selections for equi- LC n -valued mappings In this section, to each Φ : X Y we associate the mapping Φ : X → F ( Y )defined by Φ( x ) = Φ( x ), x ∈ X . Moreover, for a pair of mappings Φ , Ψ : X Y ,we will use Φ ∧ Ψ to denote their intersection, i.e. the mapping which assigns toeach x ∈ X the set [Φ ∧ Ψ]( x ) = Φ( x ) ∩ Ψ( x ). Finally, to each ε > X Y in a metric space ( Y, d ), we will associate the mapping O [Φ , ε ] : X Y defined by(5.1) O [Φ , ε ]( x ) = O ε (Φ( x )) , x ∈ X .This convention will be also used in an obvious manner for usual maps f : X → Y considering f as the singleton-valued mapping x → { f ( x ) } , x ∈ X . In these terms,for maps f, g : X → Y and ε, µ >
0, we have that f is a µ -selection for Φ : X Y with d ( f ( x ) , g ( x )) < ε for every x ∈ X , if and only if f is a selection for themapping O [Φ , µ ] ∧ O [ g, ε ].The following two constructions are due to Michael, see [11, Lemma 11.3] and[11, Proof that Lemma 5.1 implies Theorem 4.1, page 569]. They reduce theselection problem for l.s.c. mappings to that of lower locally constant mappings.For completeness, we sketch their proofs following the original arguments in [11]. Proposition 5.1.
Let ( Y, d ) be a metric space, Φ : X Y be l.s.c. and ε > .Then the mapping O [Φ , ε ] : X Y is lower locally constant.Proof. Take x ∈ X and a compact set K ⊂ O [Φ , ε ]( x ) = O ε (Φ( x )). Then K ⊂ O δ ( S ) for some finite subset S ⊂ Φ( x ) and some δ > δ < ε . SinceΦ is l.s.c., U = T y ∈ S Φ − [ O ε − δ ( y )] is an open set containing x . Moreover, x ∈ U implies S ⊂ O ε − δ (Φ( x )) and, therefore, K ⊂ O δ ( S ) ⊂ O ε (Φ( x )) = O [Φ , ε ]( x ). (cid:3) Proposition 5.2.
Let ( Y, d ) be a complete metric space, ξ : (0 , + ∞ ) → (0 , + ∞ ) be a function with ξ ( ε ) ≤ ε , and Φ : X Y be a mapping such that for each con-tinuous ξ ( ε ) -selection g : X → Y for Φ and µ > , then mapping O [Φ , µ ] ∧ O [ g, ε ] has a continuous selection. Then for every continuous ξ ( ε/ -selection g : X → Y for Φ , the mapping Φ ∧ O [ g, ε ] also has a continuous selection. ONSTRUCTING SELECTIONS STEPWISE OVER SKELETONS OF NERVES 13
Proof.
Let f = g : X → Y be a continuous ξ (2 − ε )-selection for Φ. By conditionwith µ = ξ (2 − ε ), the mapping O [Φ , ξ (2 − ε )] ∧ O [ f , − ε ] has a continuous selec-tion f : X → Y . Thus, by induction, there exists a sequence of continuous maps f n : X → E such that f n +1 is a selection for O (cid:2) Φ , ξ (cid:0) − ( n +1) ε (cid:1)(cid:3) ∧ O [ f n , − n ε ],for every n ∈ N . Then the sequence { f n : n ∈ N } is uniformly Cauchy because d ( f n +1 ( x ) , f n ( x )) < − n ε , x ∈ X . Hence, it converges uniformly to some contin-uous map f : X → Y because ( Y, d ) is complete. Since ξ (2 − n ε ) ≤ − n ε , each f n is a 2 − n ε -selection for Φ being a selection for O [Φ , ξ (2 − n ε )], see (5.1). Hence, d ( f ( x ) , Φ( x )) = 0, for each x ∈ X . Finally, we also have that d ( f ( x ) , g ( x )) ≤ ∞ X n =1 d ( f n +1 ( x ) , f n ( x )) < ∞ X n =1 − n ε = ε, x ∈ X. (cid:3) Let n ≥ −
1. A family S of subsets of a metric space ( Y, d ) is called uniformlyequi- LC n [11] if for every ε > δ ( ε ) > S ∈ S ,every continuous map of the k -sphere ( k ≤ n ) in S of diameter < δ ( ε ) can beextended to continuous map of the ( k + 1)-ball into a subset of S of diameter < ε .Just as in the case of equi- LC n families, a family S is uniformly equi- LC − iff itconsists of nonempty sets. For such a family S , by replacing δ ( ε ) with δ ( ε )2 , we getthat S is uniformly equi- LC n if there exists a function δ : (0 , + ∞ ) → (0 , + ∞ )such that(5.2) O δ ( ε ) ( y ) ∩ S k ֒ → O ε ( y ) ∩ S, for every S ∈ S , y ∈ Y and 0 ≤ k ≤ n .Evidently, we may further assume that δ ( ε ) ≤ ε , for every ε >
0. Based on thisand the results of the previous section, we now have the following two applicationsof Theorem 3.1. The first one gives a simplified proof of [11, Theorem 4.1].
Theorem 5.3.
Let E be a Banach space and S be a uniformly equi- LC n familyof subsets of E . Then there exists a function γ : (0 , + ∞ ) → (0 , + ∞ ) with thefollowing property : If X is a paracompact space with dim( X ) ≤ n + 1 , Φ : X → S is l.s.c. and g : X → E is a continuous γ ( ε ) -selection for Φ , then Φ ∧ O [ g, ε ] hasa continuous selection.Proof. Let δ ( ε ) ≤ ε be as in (5.2) with respect to the family S . Also, let λ ( ε, µ )and η ( ε ) be as in (4.3) applied to this particular function δ ( ε ). Next, definefunctions η k ( ε ) and λ k ( ε, µ ), 0 ≤ k ≤ n + 1, by(5.3) ( η n +1 ( ε ) = ε and η k ( ε ) = η (cid:0) η k +1 ( ε ) (cid:1) λ n +1 ( ε, µ ) = µ and λ k ( ε, µ ) = λ (cid:0) η k +1 ( ε ) , λ k +1 ( ε, µ ) (cid:1) . Then γ ( ε ) = η ( ε/
2) is as required. Indeed, let X and Φ be as in the theorem.Applying Proposition 5.2 with ξ ( ε ) = η ( ε ), it will be now sufficient to show thatfor every µ > η ( ε )-selection g : X → E for Φ, the mapping O [Φ , µ ] ∧ O [ g, ε ] has a continuous selection. To this end, for every 0 ≤ k ≤ n + 1, let ϕ k = O [Φ , λ k ( ε, µ )] ∧ O [ g, η k ( ε )]. According to Proposition 5.1, each ϕ k islower locally constant. Moreover, the resulting sequence of mappings ϕ k : X E ,0 ≤ k ≤ n + 1, is aspherical because by (5.2) and Corollary 4.4, ϕ k ( x ) = O λ k ( ε,µ ) (Φ( x )) ∩ O η k ( ε ) ( g ( x ))= O λ ( η k +1 ( ε ) ,λ k +1 ( ε,µ )) (Φ( x )) ∩ O η ( η k +1 ( ε )) ( g ( x )) k ֒ → O λ k +1 ( ε,µ ) (Φ( x )) ∩ O η k +1 ( ε ) ( g ( x )) = ϕ k +1 ( x ) , k ≤ n. Hence by Theorem 3.1, ϕ n +1 = O [Φ , λ n +1 ( ε, µ )] ∧ O [ g, η n +1 ( ε )] = O [Φ , µ ] ∧ O [ g, ε ]has a continuous selection. The proof is complete. (cid:3) Theorem 5.4.
Let X be a paracompact space with dim( X ) ≤ n +1 , E be a Banachspace, and Φ k : X E , ≤ k ≤ n + 1 , be a sequence of l.s.c. mappings such that { Φ k ( x ) : x ∈ X } is uniformly equi- LC k and Φ k k ֒ → Φ k +1 for every k ≤ n . Then Φ n +1 has a continuous ε -selection, for every ε > .Proof. According to (5.2) and Corollary 4.3, for each 0 ≤ k ≤ n there exists afunction δ k : (0 , + ∞ ) → (0 , + ∞ ) such that(5.4) O δ k ( ε ) (Φ k ( x )) k ֒ → O ε (Φ k +1 ( x )) , x ∈ X. Next, define functions γ k : (0 , + ∞ ) → (0 , + ∞ ), 0 ≤ k ≤ n + 1, by(5.5) γ n +1 ( ε ) = ε and γ k ( ε ) = δ k ( γ k +1 ( ε )) , k ≤ n. Finally, define a sequence of mappings ϕ k : X E by ϕ k = O [Φ k , γ k ( ε )]. It nowfollows from (5.4) and (5.5) that ϕ k ( x ) = O γ k ( ε ) (Φ k ( x )) = O δ k ( γ k +1 ( ε )) (Φ k ( x )) k ֒ → O γ k +1 ( ε ) (Φ k +1 ( x )) = ϕ k +1 ( x ) , k ≤ n. Hence, the mappings ϕ k , 0 ≤ k ≤ n + 1, form an aspherical sequence. Moreover,by Proposition 5.1, each ϕ k is lower locally constant. Since dim( X ) ≤ n + 1, byTheorem 3.1, ϕ n +1 = O [Φ n +1 , γ n +1 ( ε )] = O [Φ n +1 , ε ] has a continuous selection,i.e. Φ n +1 has a continuous ε -selection. (cid:3) We are also ready for the proof of Theorem 1.1.
Proof of Theorem 1.1.
Let X , Y , S ⊂ F ( Y ) and Φ : X → S be as in thattheorem. Since S is equi- LC n , by [5, Theorem 1] (see, also, [11, Proposition 2.1]), S S can be embedded into a Banach space E such that S ⊂ F ( E ) is uniformlyequi- LC n . Then by Theorem 5.4, applied with Φ k = Φ, 0 ≤ k ≤ n + 1, themapping Φ has a continuous ε -selection, for every ε >
0. Hence, by Theorem 5.3,Φ = Φ has a continuous selection as well. (cid:3)
ONSTRUCTING SELECTIONS STEPWISE OVER SKELETONS OF NERVES 15
Another application of Theorems 5.3 and 5.4 is the following generalisation ofTheorem 1.1, see [14], [12, Theorem 7.2] and [9, Corollary 7.10].
Corollary 5.5.
Let X be a paracompact space with dim( X ) ≤ n + 1 , Y be a com-pletely metrizable space, and Φ k : X → F ( Y ) , ≤ k ≤ n +1 , be a sequence of l.s.c.mappings such that Φ k k ֒ → Φ k +1 for k ≤ n , while each family { Φ k ( x ) : x ∈ X } , for k ≤ n + 1 , is equi- LC k . Then Φ n +1 has a continuous selection.Proof. As before, the proof is reduced to the case when Y = E is a Banach space,and each family { Φ k ( x ) : x ∈ X } ⊂ F ( E ), 0 ≤ k ≤ n + 1, is uniformly equi- LC k .Then by Theorem 5.4, Φ n +1 has a continuous ε -selection, for every ε >
0. Finally,by Theorem 5.3, Φ n +1 also has a continuous selection. (cid:3) References [1] S. M. Ageev,
A nonpolyhedral proof of the Michael finite-dimensional selection theorem ,Fundam. Prikl. Mat. (2005), no. 4, 3–22, (in Russian).[2] A. Chigogidze and V. Valov, Extensional dimension and C -spaces , Bull. London Math. Soc. (2002), no. 6, 708–716.[3] G. De Marco and R. G. Wilson, Realcompactness and partitions of unity , Proc. Amer. Math.Soc. (1971), 189–194.[4] C. H. Dowker, Mappings theorems for non-compact spaces , Amer. J. Math. (1947),200–242.[5] J. Dugundji and E. Michael, On local and uniformly local topological properties , Proc. Amer.Math. Soc. (1956), 304–307.[6] R. Engelking, General topology, revised and completed edition , Heldermann Verlag, Berlin,1989.[7] V. V. Fedorchuk and V. V. Filippov,
General topology. Basic constructions , Fizmatlit,Moscow, 2006 (in Russian).[8] V. Gutev,
Factorizations of set-valued mappings with separable range , Comment. Math.Univ. Carolin. (1996), no. 4, 809–814.[9] , Selections and approximations in finite-dimensional spaces , Topology Appl. (2005), 353–383.[10] S. Lefschetz,
Algebraic Topology , American Mathematical Society Colloquium Publications,v. 27, American Mathematical Society, New York, 1942.[11] E. Michael,
Continuous selections II , Ann. of Math. (1956), 562–580.[12] D. Repovˇs and P. V. Semenov, Continuous selections of multivalued mappings , Mathematicsand its applications, vol. 455, Kluwer Academic Publishers, The Netherlands, 1998.[13] K. Sakai,
Geometric aspects of general topology , Springer Monographs in Mathematics,Springer, Tokyo, 2013.[14] E. V. Schepin and N. B. Brodsky,
Selections of filtered multivalued mappings , Trudy Mat.Inst. Steklova (1996), 220–240, (in Russian; Engl. Transl. in: Proc. Steklov Inst. Math.212 (1996), 218–239.).[15] P. Urysohn, ¨Uber die M¨achtigkeit der zusammenh¨angenden Mengen , Math. Ann. (1925),no. 1, 262–295.[16] V. Uspenskij, A selection theorem for C -spaces , Topology Appl. (1998), 351–374.[17] V. Valov, Continuous selections and finite C -spaces , Set-Valued Analysis (2002), no. 1,37–51. [18] J. H. C. Whitehead, Simplicial Spaces, Nuclei and m-Groups , Proc. London Math. Soc. (2) (1939), no. 4, 243–327.[19] , Combinatorial homotopy. I , Bull. Amer. Math. Soc. (1949), 213–245. Department of Mathematics, Faculty of Science, University of Malta, MsidaMSD 2080, Malta
E-mail address ::