COMPLETE METRIC ABSOLUTE NEIGHBORHOOD RETRACTS

WIESŁAW KUBIŚ

Abstract. We characterize complete metric absolute (neighborhood) retracts in terms of existence of certain maps of CW-polytopes. Using our result, we prove that a compact metric space with a convex and locally convex simplicial structure is an AR. This answers a question of Kulpa from [5]. As another application, we prove that the hyperspace of closed subsets of a separable Banach space endowed with the Wijsman topology is an absolute retract.

1. Introduction

A metrizable space X is an absolute (neighborhood) retract (briefly: AR (ANR)) if it is a retract of (an open subset of) a normed linear space containing X as a closed subset. There are several known characterizations of ANR’s stated in terms of maps of CW-polytopes. Probably the most well-known is Dugundji-Lefschetz’ theorem about realizations of polytopes. Another result in this spirit is due to Nhu [7]. We introduce a metric property (Property (B) below) which, roughly speaking, says that there is a sequence of maps of CW-polytopes with some ‘compatibility’ conditions, related to the metric. We prove (Section 2) that a complete metric space with this property is an ANR; a stronger version of Property (B) (called Property (B^{∗})) implies that the space is an AR. It appears that Property (B) characterizes ANR’s among complete metric spaces; we also give an example of a (non-complete) metric space with Property (B), which is not an ANR. Property (B) does not require the existence of extensions of any maps, which is required in Dugundji-Lefschetz’ characterization. In Section 3 we show that the realization property of Dugundji-Lefschetz implies Property (B). This gives a proof of Dugundji-Lefschetz’ theorem in the case of completely metrizable spaces. The last section is devoted to applications. First we consider simplicial structures introduced by Kulpa [5] and we show that a compact metric space with a convex and locally convex simplicial structure is an AR. This solves Kulpa’s problem from [5]. As a second application, we study hyperspaces of closed sets endowed with the Wijsman topology. This topology is important and useful in the analysis of set convergence and in optimization theory; for references see Beer’s book [1]. We show that if a given metric space has the property that after removing finitely many closed balls, the remaining part is path-wise connected, then its Wijsman hyperspace has Property (B^{∗}). Consequently, the Wijsman hyperspace of a separable Banach space is an absolute retract.

1991 Mathematics Subject Classification. Primary: 54C55, 54E50, Secondary: 54C20, 54B20.

Key words and phrases. absolute (neighborhood) retract, complete metric space, polytope, the Wijsman topology.

1.1. Notation. We denote by [X]^{<ω} and [X]^{n} the collection of all finite and n-element subsets of X respectively; ω denotes the set of all nonnegative integers. Given any set S we shall denote by Σ(S) the union of all geometric simplices with vertices in S, endowed with the CW-topology. More precisely, Σ(S) is the set of all formal convex combinations of the form ∑_{s∈S} λ_{s}s, where λ_{s} = 0 for all but finitely many s ∈ S; a subset U ⊂ Σ(S) is open if its intersection with any simplex σ of Σ(S) is open in σ (with respect to the standard topology on σ). When S is finite, Σ(S) is called the geometric or abstract simplex with the set of vertices S. A subsimplex (or a face) of a simplex Σ(S) is, by definition, a simplex Σ(T), where T ⊂ S. The boundary of a simplex σ = Σ(S) is bd σ = ⋃_{T ⊂S, T ≠S} Σ(T). An abstract polytope is a space of the form P = ⋃_{T ∈A} Σ(T), where A is any family of sets, endowed with the CW-topology, i.e. P is a subspace of Σ(S), where S = ⋃ A. S is the set of vertices of P and we write S = vert P. Observe that P can also be written as ⋃_{T ∈A_{0}} Σ(T), where A_{0} = {T: (∃ T′ ∈ A) T ∈ [T′]^{<ω}}. A subpolytope of an abstract polytope P = ⋃_{T ∈A} Σ(T) is a polytope Q = ⋃_{R∈B} Σ(R) such that every simplex of Q (i.e. a face of Σ(R) for some R ∈ B) is also a simplex of P, i.e. for every R ∈ B there exists T ∈ A with R ⊂ T. A polytope P is convex if P = ⋃_{T ∈[S]^{<ω}} Σ(T), where S = vert P. Clearly, this agrees with the definition of the convex hull of S, when we consider a polytope as a subset of a real linear space. By a polytope in a topological space Y we mean a continuous map φ: P → Y of an abstract polytope P. We say that S = vert P is the set of vertices of φ and we write S = vert φ. Sometimes φ is called a singular polytope in Y or a realization of P in Y. We define the notion of subpolytope, convex polytope and simplex like in the abstract case.

2. Main result

Let U be a collection of subsets of a topological space Y. We write A ≺ U whenever A is a set contained in some element of U. A polytope φ in Y is U-dense if U ∩ φ[vert φ] ≠ ∅ whenever U ∈ U \ {∅}. Let U, V be two open covers of Y. We say that a polytope φ is (U, V)-compatible if for every finite set S ⊂ vert φ we have Σ(S) ⊂ dom φ and φ[Σ(S)] ≺ V whenever φ[S] ≺ U. By mesh U we mean the supremum of diameters of members of U. We say that a metric space Y has Property (B) provided there exists a sequence of open covers {U_{n}}_{n∈ω} satisfying the following conditions:

(a) for each n ∈ ω, U_{n+1} is a star-refinement of U_{n},

(b) ∑_{n∈ω} mesh U_{n} < +∞,

(c) for each n ∈ ω there is m > n + 5 and there exists a U_{m+1}-dense polytope in Y, which is simultaneously (U_{m}, U_{n+5})- and (U_{n+1}, U_{n})-compatible.

If additionally, for some n ∈ ω there is a convex polytope satisfying (c) then we say that Y has Property (B^{∗}).

Theorem 1. Every complete metric space with Property (B) is an absolute neighborhood retract. A complete metric space with Property (B^{∗}) is an absolute retract.

Proof. We start with two lemmas. We assume here that Y is a complete metric space with Property (B), A is a closed subset of a metrizable space X and f: A → Y is a fixed continuous map.

Lemma 1. Let n > 0 and suppose that g: X → Y is a continuous map such that g|A is U_{n+3}-close to f. Then there exists a continuous map g′: X → Y which is U_{n−1}-close to g and U_{n+4}-close to f on A.

Proof. Let m > n + 5 be as in condition (c) of Property (B). Set U = U_{m+1}. For U ∈ U define

U^{∗} = f^{−1}[U] ∪ (g^{−1}[star(U, U_{n+3})] \ A).

Observe that {U^{∗}}_{U∈U} is an open cover of X. Let {h_{U}}_{U∈U} be a locally finite partition of unity such that h_{U}^{−1}[(0, 1]] ⊂ U^{∗} for U ∈ U. By condition (c) of Property (B), there exists a U-dense polytope φ in Y which is simultaneously (U_{m}, U_{n+5})- and (U_{n+1}, U_{n})- compatible. For each U ∈ U \ {∅} choose y_{U} ∈ vert φ such that φ(y_{U}) ∈ U. Fix t ∈ X and consider U_{t} = {U ∈ U: h_{U}(t) > 0}. Then g(t) ∈ star(U, U_{n+3}) for U ∈ U_{t}. Let S_{t} = {y_{U}: U ∈ U_{t}}. Then φ[S_{t}] ⊂ star(g(t), U_{n+2}) ≺ U_{n+1} and hence Σ(S_{t}) ⊂ dom φ. Define a map g′: X → Y by setting

g′(t) = φ(∑_{U∈U} h_{U}(t)y_{U}).

Clearly g′ is continuous and U_{n−1}-close to g since {g′(t)}∪φ[S_{t}] ≺ U_{n} and {g(t)}∪φ[S_{t}] ≺ U_{n+1}. Suppose now that t ∈ A. Then f(t) ∈ ⋂ U_{t} and consequently φ[S_{t}]∪{f(t)} ⊂ star(f(t), U) ≺ U_{m}. Thus φ[S_{t}] ∪ {g′(t)} ∈ φ[Σ(S_{t})] ≺ U_{n+5} which means that g′|A is U_{n+4}-close to f. □

Lemma 2. There exists an open set W ⊃ A and a continuous map g: W → Y which is U_{4}-close to f on A. If Y has Property (B^{∗}) then we may assume that W = X.

Proof. Applying condition (c) of Property (B) (for n = 0) we get m > 5 and a polytope φ which is U_{m+1}-dense and (U_{m}, U_{5})-compatible. Set U = U_{m+1}. By paracompactness, there is a locally finite open cover {H_{U}}_{U∈U} of X such that A ∩ cl H_{U} ⊂ f^{−1}[U] for every U ∈ U. Set

V_{U} = H_{U} \ ⋃{cl H_{G}: G ∈ U & A ∩ cl H_{G} ∩ H_{U} = ∅}.

Observe that each V_{U} is open in X, A ⊂ ⋃_{U∈U} V_{U} and V_{U_{1}} ∩ V_{U_{2}} ≠ ∅ implies U_{1} ∩ U_{2} ≠ ∅. The last property follows from the fact that if V_{U_{1}} ∩V_{U_{2}} ≠ ∅ then there is t ∈ A∩cl H_{U_{1}} ∩ H_{U_{2}} and consequently f(t) ∈ U_{1} ∩ U_{2}. Let W = ⋃_{U∈U} V_{U} and let {h_{U}}_{U∈U} be a locally finite partition of unity in W such that h_{U}^{−1}[(0, 1]] ⊂ V_{U} for every U ∈ U. Now, for each U ∈ U \ {∅} choose y_{U} ∈ vert φ so that φ(y_{U}) ∈ U. Define

(∗)  g(t) = φ(∑_{U∈U} h_{U}(t)y_{U}),  t ∈ W.

Observe that g is well-defined, since if U_{t} = {U ∈ U: h_{U}(t) > 0} then {φ(y_{U}): U ∈ U_{t}} ⊂ star(U_{0}, U), where U_{0} ∈ U_{t} is arbitrary (because U_{1} ∩ U_{2} ≠ ∅ whenever U_{1}, U_{2} ∈ U_{t}) and consequently Σ({y_{U}: U ∈ U_{t}}) ⊂ dom φ. As in the proof of the previous Lemma, one can check that g|A is U_{4}-close to f. Finally, if Y has Property (B^{∗}) then we may assume that φ is a convex polytope, so formula (∗) well defines a continuous map on the entire space X. Thus, in this case we can set W = X. □

Theorem 1 follows immediately from Lemma 1 and Lemma 2. Indeed, using Lemma 2 we get a continuous map g_{0} : W → Y which is U_{4}-close to f, where W ⊃ A is open. If Y has Property (B^{∗}) then W = X. Now we can use inductively Lemma 1 to obtain a sequence of continuous maps g_{n}: W → Y such that g_{n+1} is U_{n−1}-close to g_{n} and U_{n+4}-close to f on A. By condition (b) of Property (B), the sequence {g_{n}}_{n∈ω} converges uniformly to a continuous map f′: W → Y which is an extension of f (here we have used the completeness of Y). □

We now show that every metric ANR/AR has Property (B)/(B^{∗}).

Proposition 1. Let Y be a metric ANR. Then there exists a polytope φ in Y with vert φ = Y and there exists a sequence {U_{n}}_{n∈ω} of open covers of Y such that for each n ∈ ω, mesh U_{n} ⩽ 2^{−n}, U_{n+1} is a star-refinement of U_{n} and φ is (U_{n+1}, U_{n})-compatible. If additionally, Y is an AR then φ is a convex polytope.

Proof. By the theorem of Arens-Eells we can assume that Y is a closed subset of a normed linear space E. Let r: W → Y be a retraction, where W ⊃ Y is open in E. Define

P = ⋃{Σ(S): S ∈ [Y]^{<ω} & conv_{E} S ⊂ W } ⊂ Σ(Y).

Let ψ: P → E be the unique affine map with ψ|Y = id_{Y}. Then φ = rψ is a polytope in Y with vert φ = Y. Let U_{0} be any open cover of Y with mesh ⩽ 1. Suppose that covers U_{0}, …, U_{n} are already defined so that mesh U_{i} < 2^{−(i+1)} and they satisfy conditions (a) and (b). By the continuity of r, there exists an open cover V of W, consisting of convex sets and such that {r[V]: V ∈ V} is a refinement of U_{n}. Now let U_{n+1} be a star-refinement of U_{n} with mesh ⩽ 2^{−(n+1)}, which is also a refinement of V. Then φ is (U_{n+1}, U_{n})-compatible. Finally, if Y is an AR then W = E and hence P = Σ(Y). □

Below we describe an example of a separable metric space with Property (B^{∗}), which is not an ANR. Thus, the completeness assumption in Theorem 1 is essential.

Example 1. Consider the Hilbert cube Q = [0, 1]^{ω} endowed with the product metric. There exists a sequence {A_{n}}_{n∈ω} of pairwise disjoint dense convex subsets of Q. Indeed, if {B_{n}}_{n∈ω} is a decomposition of ω into infinite sets then we can set

A_{n} = {x ∈ Q: ∃ i ∈ B_{n} (x(i) > 0 & (∀ j > i) x(j) = 0)}.

Now, for each n ∈ ω choose finite D_{n} ⊂ A_{n} which is 1/n-dense in Q and define Y = ⋃_{n∈ω} conv D_{n}. Clearly, Y is dense in Q, so Q is the completion of Y. Let {U_{n}}_{n∈ω} be a sequence of finite open covers of Y such that mesh U_{n} ⩽ 2^{−n}, U_{n+1} is a star-refinement of U_{n} and each element of U_{n} is of the form U ∩ Y, where U ⊂ Q is convex. Let φ_{k}: Σ(D_{k}) → conv D_{k} ⊂ Y be the unique affine map which extends id_{D_{k}}.

Then φ_{k} is (U_{n}, U_{n})-compatible for each n ∈ ω; moreover φ_{k} is U_{n}-dense for a sufficiently large k. It follows that Y has Property (B^{∗}). On the other hand, Y is not an ANR, since it is not locally path-wise connected at any point: as no continuum can be decomposed into countably many nonempty closed subsets, every path in Y is contained in conv D_{n} for some n, but these sets are pairwise disjoint and nowhere dense in Y.

3. A relation to Dugundji-Lefschetz’ theorem

We recall the theorem of Lefschetz [6] and Dugundji [4] characterizing metric ANR’s, stated in terms of realizations of polytopes. Let P be a CW-polytope and let Q be a subpolytope of P. A continuous map φ: Q → Y is a partial realization of P relative to a cover U, provided Q contains all the vertices of P and for each simplex σ of P, φ[Q ∩ σ] ≺ U. If Q = P then φ is a full realization relative to U. Dugundji-Lefschetz’ theorem says that a metrizable space Y is an ANR if and only if every open cover U of Y has an open refinement S(U) such that for every CW-polytope P, every partial realization of P relative to S(U) can be extended to a full realization of P relative to U.

We show that every metric space with the realization property stated above, has Property (B). This provides a proof of the ”if” part of Dugundji-Lefschetz’ theorem, in the case of completely metrizable spaces.

Fix a metric space Y with the above realization property. Let U_{0} be any open cover of Y with finite mesh and, inductively, let U_{n+1} be an open star-refinement of S(U_{n}) with mesh ⩽ 2^{−n}. Clearly, the sequence {U_{n}}_{n∈ω} satisfies conditions (a), (b) of Property (B). We check (c). Fix n ∈ ω and set m = n+6. Choose any U_{m+1}-dense set S ⊂ Y. Consider

P_{1} = ⋃{Σ(T): T ∈ [S]^{<ω} & T ≺ U_{m}}.

Clearly, P_{1} is a CW-polytope, S is a subpolytope of P_{1} and the identity map id_{S}: S → Y is a partial realization of P_{1} relative to U_{m}. As U_{m} is a refinement of S(U_{n+5}), there exists a full realization φ_{1} : P_{1} → Y of P_{1} relative to U_{n+5} with φ_{1}|S = id_{S}. Observe that φ_{1} is (U_{m}, U_{n+5})-compatible. Define

P_{2} = ⋃{Σ(T): T ∈ [S]^{<ω} & φ_{1}[Σ(T) ∩ P_{1}] ≺ S(U_{n})}.

Then P_{1} is a subpolytope of P_{2} and φ_{1} is a partial realization of P_{2} relative to S(U_{n}). Let φ_{2} : P_{2} → Y be a full realization of P_{2} relative to U_{n} which extends φ_{1}. Clearly, φ_{2} is (U_{m}, U_{n+5})-compatible. Fix T ∈ [S]^{<ω} and U ∈ U_{n+1} with T ⊂ U. Set Q = Σ(T) ∩ P_{1}. For each face σ of Σ(T) with σ ⊂ Q there is W_{σ} ∈ U_{n+5} with φ_{1}[σ] ⊂ W_{σ}. We have

φ_{1}[Q] ⊂ U ∪ ⋃{W_{σ}: σ is a face of Σ(T) with σ ⊂ Q} ⊂ star(U, U_{n+5}),

thus φ_{1}[Q] ≺ S(U_{n}). Hence Σ(T) is a simplex in P_{2} and φ_{2}[Σ(T)] ≺ U_{n}. It follows that φ_{2} is (U_{n+1}, U_{n})-compatible. This shows that condition (c) of Property (B) is satisfied.

4. Applications

4.1. Simplicial structures. Following Kulpa [5] we say that a collection F consisting of simplices in a space Y is a simplicial structure in Y provided σ ∈ F implies that vert σ ⊂ Y, σ| vert σ = id_{vert σ} and every subsimplex of σ is in F. The pair (Y, F) is then called a simplicial space. We write vert F = {vert σ: σ ∈ F}. A simplicial space (Y, F) is locally convex if for each p ∈ Y and its neighborhood V there exists a smaller neighborhood U of p such that [U]^{<ω} ⊂ vert F and for every σ ∈ F, vert σ ⊂ U implies im σ := σ[Σ(vert σ)] ⊂ V. A simplicial space (Y, F) is convex if every finite subset of Y is in vert F. A theorem of Kulpa [5] says that every convex locally convex simplicial space has the fixed point property for continuous self-maps with compact images. We show that every compact metric space with such a property is an AR. This answers a question posed by Kulpa in [5].

Theorem 2. Every compact metric space with a convex and locally convex simplicial structure is an AR.

Proof. Fix an open cover U of a compact metric space Y with a convex, locally convex simplicial structure F. Denote by R(U) a fixed refinement V of U with the following property:

(∀ V ∈ V)(∃ U ∈ U)(∀ σ ∈ F) vert σ ⊂ V ⟹ im σ ⊂ U.

Now define a sequence of open covers U_{n} such that U_{n+1} is a finite star-refinement of R(U_{n}) with mesh ⩽ 2^{−n}. Clearly, the sequence {U_{n}}_{n∈ω} satisfies conditions (a) and (b) of Property (B^{∗}). We check condition (c). Fix n ∈ ω and let m = n + 6. There exists a U_{m+1}-dense simplex σ ∈ F, since F is convex and Y is compact. Observe that σ is (U_{k+1}, U_{k})-compatible for each k ∈ ω. Indeed, if S ⊂ vert σ and S ≺ U_{k+1} then S ≺ R(U_{k}) so σ[Σ(S)] ≺ U_{k}. This shows that Y has Property (B^{∗}). By Theorem 1, Y is an AR. □

4.2. Hyperspaces. For a topological space X we denote by CL(X) the hyperspace of all nonempty closed subsets. We write T_{V} for the Vietoris topology on CL(X). Let (X, d) be a metric space. The Wijsman topology is the least topology T_{W_{d}} on CL(X) such that for each p ∈ X the function dist(p, ·): CL(X) → ℝ is continuous. Equivalently, T_{W_{d}} is the topology generated by all sets of the form:

U^{−}(p, r) = {A ∈ CL(X): dist(p, A) < r},
U^{+}(p, r) = {A ∈ CL(X): dist(p, A) > r},

where p ∈ X and r > 0. The Wijsman topology is weaker than the Vietoris one. Also, (CL(X), T_{W_{d}}) is metrizable (Polish) iff (X, d) is separable (Polish) (Beer-Costantini’s theorem, see [3]). For a survey on hyperspace topologies we refer to Beer’s book [1].

Theorem 3. Let (X, d) be a Polish space with the following property:

(∗) if K is a finite family of closed balls in X then X \ ⋃ K is path-wise connected.

Then (CL(X), T_{W_{d}}) is an absolute retract.

It is clear that to divide ℝ^{n} we need at least n + 1 compact convex sets. It follows that a finite union of bounded closed convex sets in an infinite dimensional normed space does not divide the space. Hence, applying Theorem 3, we get the following.

Corollary 1. Let (X, ∥ · ∥) be an infinite-dimensional separable Banach space. Then (CL(X), T_{W_{∥·∥}}) is an absolute retract.

It has been proved by Sakai & Yang [8] that the Wijsman hyperspace of ℝ^{n} is homeomorphic to the Hilbert cube minus a point (the authors of [8] consider hyperspaces with the Fell topology which, in the case of locally compact metric spaces, is equivalent to the Wijsman one). So the Wijsman hyperspace of every separable Banach space is an AR.

Proof of Theorem 3. Fix a Polish space (X, d) with property (∗). Denote by B the collection of all sets of the form U^{−}(p_{1}, r_{1})∩⋯∩U^{−}(p_{k}, r_{k})∩U^{+}(q_{1}, s_{1})∩⋯∩U^{+}(q_{l}, s_{l}), where p_{1}, …, p_{k}, q_{1}, …, q_{l} ∈ X, r_{1}, …, r_{k}, s_{1}, …, s_{l} > 0. Clearly, B is an open base for T_{W_{d}}. The following two lemmas refer to the Wijsman topology on CL(X).

Lemma 3. For each W ∈ B the set [X]^{<ω} ∩ W is path-wise connected.

Proof. Let W = U^{−}(p_{1}, r_{1})∩⋯∩U^{−}(p_{k}, r_{k})∩U^{+}(q_{1}, s_{1})∩⋯∩U^{+}(q_{l}, s_{l}), where p_{i}, q_{i}, r_{i}, s_{i} are as above, and denote G = X \ (\overline{B}(q_{1}, s_{1}) ∪ ⋯ ∪ \overline{B}(q_{l}, s_{l})), where \overline{B}(q, s) is the closed ball centered at q ∈ X with radius s > 0. By (∗), G is path-wise connected. Fix a, b ∈ [X]^{<ω} ∩ W. For each (x, y) ∈ a × b choose a path γ_{x,y}: [0, 1] → G with γ_{x,y}(0) = x and γ_{x,y}(1) = y. Define Γ: [0, 1] → CL(X) by

Γ(t) = \begin{cases}⋃_{(x,y)∈a×b}\{x, γ_{x,y}(2t)\}, & t ⩽ 1/2, \\ ⋃_{(x,y)∈a×b}\{γ_{x,y}(2 − 2t), y\}, & t ⩾ 1/2. \end{cases}

Clearly, Γ(t) ∈ W for every t ∈ [0, 1] and Γ(0) = a and Γ(1) = b. A routine verification shows that Γ is continuous (it is actually continuous with respect to the Vietoris topology). □

Lemma 4. Let φ: bd σ → CL(X) be a continuous map from the boundary of a geometric simplex σ. If the dimension of σ is at least 2 then there exists a continuous extension ψ: σ → CL(X) of φ such that for each W ∈ B we have ψ[σ] ⊂ W whenever φ[bd σ] ⊂ W.

Proof. Take a Vietoris continuous map r: σ → CL(bd σ) which extends the natural injection i: bd σ → CL(bd σ) (see [2, Lemma 3.3]). Define

ψ(s) = cl_{X} ⋃ φ[r(s)], s ∈ σ.

An easy verification shows that ψ is continuous. Clearly, ψ is an extension of φ. If φ[bd σ] ⊂ U^{−}(p, r) then also ψ[σ] ⊂ U^{−}(p, r). If φ[bd σ] ⊂ U^{+}(p, r) then for s ∈ σ we have dist(p, ψ(s)) ⩾ r and, using the compactness of r(s) and the continuity of dist(p, φ( ·)), we get dist(p, ψ(s)) > r. Thus also ψ[σ] ⊂ U^{+}(p, r). □

Fix a complete metric ϱ in (CL(X), T_{W_{d}}). We will show that (CL(X), ϱ) has Property (B^{∗}). Let {U_{n}}_{n∈ω} be a sequence of covers of CL(X) such that for each n ∈ ω, U_{n} ⊂ B, mesh U_{n} ⩽ 2^{−n} and U_{n+1} is a star-refinement of U_{n}. We show that condition (c) of Property (B^{∗}) is fulfilled. Fix n ∈ ω and set m = n + 6. As [X]^{<ω} is dense in (CL(X), T_{W_{d}}), we can find a set S ⊂ [X]^{<ω} which is U_{m+1}-dense. Define

P_{1} = ⋃{Σ(T): T ∈ [S]^{<ω} & T ≺ U_{m}}.

Denote by P_{1}^{(1)} the 1-skeleton of P_{1}, i.e. the union of all at most 1-dimensional simplices of P_{1}. By Lemma 3, the identity map id: S → S can be continuously extended to φ^{1} : P_{1}^{(1)} → CL(X), such that for each T ∈ [S]^{2} we have φ^{1}[Σ(T)] ⊂ W for some W ∈ U_{m}, whenever T ≺ U_{m}. Now by Lemma 4, φ^{1} can be extended to a continuous map φ_{1} : P_{1} → CL(X), which is a (U_{m}, U_{n+5})-compatible polytope. Next define

P_{2} = ⋃{Σ(T): T ∈ [S]^{<ω} & T ≺ U_{n+1}}.

Then P_{1} is a subpolytope of P_{2}. Again by Lemma 3, φ_{1} can be continuously extended to φ^{2} : P_{1} ∪ P_{2}^{(1)} → CL(X) with the property analogous to φ^{1}. Finally, as φ^{2}[P_{2}^{(0)}] ⊂ [X]^{<ω} and [X]^{<ω} is path-wise connected, so again using Lemma 4, φ^{2} can be extended to a continuous map φ_{2} : Σ(S) → CL(X) which is (U_{n+1}, U_{n})-compatible. So φ_{2} is a U_{m+1}-dense convex polytope in (CL(X), T_{W_{d}}) which is both (U_{m}, U_{n+5})- and (U_{n+1}, U_{n})- compatible. This shows that (CL(X), T_{W_{d}}) has Property (B^{∗}). □

Acknowledgements. The author would like to thank the referee for many valuable remarks and comments.

References

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[2] D. Curtis, Nguyen To Nhu, Hyperspaces of finite subsets which are homeomorphic to ℵ_{0}- dimensional linear metric spaces, Topology Appl. 19 (1985) 251–260.

[3] C. Costantini, Every Wijsman topology relative to a Polish space is Polish, Proc. Amer. Math. Soc. 123 (1995) 2569-2574.

[4] J. Dugundji, Absolute neighborhood retracts and local connectedness in arbitrary metric spaces, Comp. Math. 13 (1958) 229-246.

[5] W. Kulpa, Convexity and the Brouwer fixed point theorem, Top. Proc. 22 (1997), Summer, 211-235.

[6] S. Lefschetz, On compact spaces, Ann. Math. 32 (1931) 521-538.

[7] N.T. Nhu, Investigating the ANR-property of metric spaces, Fund. Math. 124 (1984) 243-254.

[8] K. Sakai, Z. Yang, Hyperspaces of non-compact metric spaces which are homeomorphic to the Hilbert cube, Topology Appl. 127 (2003) 331–342.

University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland

E-mail address: kubis@ux2.math.us.edu.pl

Current address: Ben Gurion University of the Negev, Beer-Sheva, Israel

E-mail address: kubis@math.bgu.ac.il