# Homotopy Characterization of ANR Function Spaces

Authors: Jaka Smrekar
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

LetY be an absolute neighbourhood retract (ANR) for the class of metric spaces and let X be a topological space. Let YX denote the space of continuous maps from X to Y equipped with the compact open topology. We show that if X is a compactly generated Tychonoff space and Y is not discrete, then YX is an ANR for metric spaces if and only if X is hemicompact and YX has the homotopy type of a CW complex.


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## Body

## 1. Introduction

LetY be an absolute neighbourhood retract for metric spaces (henceforth abbreviated as “ANR”). This means that whenever Y is embedded in a metric space as a closed subspace Y^, there exists a retraction of an open neighbourhood onto Y^. We refer the reader to the first part of Mardešić [1] for a scenic survey of the theory of ANRs.LetX be a topological space. The question that this paper is concerned with is when is YX, the space of continuous functions  X→Y, equipped with the compact open topology, also an ANR. A basic result of Kuratowski (see [2, page 284]), which is a consequence of the classical homotopy extension theorem of Borsuk, states that YX is an ANR if X is a metrizable compactum.For a negative example, consider the discrete space of natural numbersℕ and the two-point discrete ANR {0,1}. Then {0,1}ℕ is a Cantor set and hence certainly not an ANR. In fact, as the path components of {0,1}ℕ are not open, it does not even have the homotopy type of a CW complex; that is, it is not homotopy equivalent to any CW complex. As every ANR has the homotopy type of a CW complex, this provides a necessary condition for YX to be an ANR.In fact, a topological space has the homotopy type of an ANR if and only if it has the homotopy type of a CW complex (see Milnor [3, Theorem 2]). However, there are numerous examples of spaces that have CW homotopy type but are not ANRs. For example, let F be the topological cone over the convergent sequence {0,1,1/2,1/3,…}. Then F is contractible; that is, it has the homotopy type of a one-point CW complex. But as F is not locally path connected, it is not an ANR.On the other hand, Cauty [4] showed that a metrizable space is an ANR if and only if each open subspace has the homotopy type of a CW complex. It turns out that the space of functions into an ANR inherits a good deal of reasonable behaviour from the target space. Thus, under mild restrictions on X, if Y is an ANR and YX is a metrizable space with the homotopy type of a CW complex, YX is in fact an ANR. Put another way, YX is an ANR if and only if Y is an ANR and YX is a metrizable space with the homotopy type of an ANR.Basic Definitions and Conventions. A topological space X is called hemicompact if X is the union of countably many of its compact subsets {Ki|i} which dominate all compact subsets in X. This means that for each compact C⊂X there exists i with C⊂Ki. (The perhaps somewhat noninformative word “hemicompact” was introduced by Arens [5] in relation to metrizability of function spaces. See the beginning of Section 2.)A spaceX is compactly generated if the compact subspaces determine its topology. That is, a subset A is closed in X if and only if A∩C is closed in C for each compact subspace C. Such spaces are also commonly called k-spaces (see, e.g., Willard [6]). We do not require a hemicompact or a compactly generated space to be Hausdorff.A spaceX is Tychonoff if it is both completely regular and Hausdorff. Locally compact Hausdorff spaces and normal Hausdorff spaces are Tychonoff (examples of the latter are all metric spaces and all CW complexes).The termsmap and continuous function will be used synonymously.A mapf:Y→Y′ is a homotopy equivalence if there exists a map g:Y′→Y (called a homotopy inverse) for which the composites g∘f and f∘g are homotopic to their respective identities. In this case, Y and Y′ are called homotopy equivalent, and we say that Y′ has the homotopy type of Y.The following are our main results.Theorem 1.
LetX be a compactly generated hemicompact space and let Y be an ANR. Then YX is an ANR if and only if YX has the homotopy type of an ANR, which is if and only if it has the homotopy type of a CW complex.We call a (not necessarily Hausdorff) space locally compact if each point is contained in the interior of a compact set. It is well-known that compactly generated spaces are precisely quotient spaces of locally compact spaces. Compactly generated hemicompact spaces seem to be important enough to warrant an analogous characterization. In the appendix, we prove that they arise as nice quotient spaces ofσ-compact locally compact spaces.Assuming additional separation properties, Theorem1 can be strengthened as follows.Corollary 2.
LetX be a compactly generated Tychonoff space and let Y be an ANR which contains an arc. Then YX is an ANR if and only if X is hemicompact and YX has the homotopy type of a CW complex.Theorem1 is a considerable extension of Theorem 1.1 of [7] where the equivalence was proved using a different technique under the more stringent requirement that X be a countable CW complex. Our proof of  Theorem 1 leans on Morita’s homotopy extension theorem for P0-embeddings (see Morita [8]).Even whenX is a countable CW complex, it is highly nontrivial to determine whether or not the function space YX has the homotopy type of a CW complex. The interested reader is referred to papers [7, 9, 10] for more on this.


## 2. Proof of  Theorem1

For subsetsA of the domain space and B of the target space, we let G(A,B) denote the set of all maps f that map the set A into the set B. For topological spaces X and Y, the standard subbasis of the compact open topology on YX is the collection 𝒫 of all G(K,V)⊂YX with K a compact subset of X and V an open subset of Y.To prove  Theorem1, we use the fact that ANRs for metric spaces are precisely the metrizable absolute neighbourhood extensors for metric spaces (abbreviated as “ANE”); see, for example, Hu [11, Theorem 3.2]. A space Y is an ANE if every continuous function A→Y, where A is a closed subspace of a metric space, extends continuously over a neighbourhood of A.Note that ifX is a hemicompact space with the sequence of “distinguished” compact sets {Ki}, the map into the countable Cartesian product
(*)YX⟶∏i=1∞YKi,definedbyf↦{f∣Ki}
is an embedding (see also Cauty [12]). Consequently, if di denotes the supmetric on YKi induced by a metric on Y, then YX is metrizable by the metric
(**)d(f,g)=∑i=1∞min{12i,di(f∣Ki,g∣Ki)}.
(See Arens [5, Theorem 7].) Given the hypotheses of  Theorem 1, therefore, we need to show that for every pair (Z,A) with Z metric and A closed in Z, every continuous function f:A→YX extends continuously over a neighbourhood of A in Z. We need some preliminary results.First, we state the classical exponential correspondence theorem with minimal hypotheses. Here, a space is regular if points can be separated from closed sets by disjoint open sets.Proposition 3.
LetX, Y, andZ be topological spaces. Let f:Z→YX be any function with set-theoretic adjoint f^:Z×X→Y. If f^ is continuous, then f is (well-defined and) continuous. For the converse, suppose that X is locally compact. If f is continuous and, in addition, X is regular or Y is regular, then f^ is continuous. This accounts for a bijection (YX)Z↔Y(X×Z).Proof.
The requirement thatX be regular is standard. (See, e.g.,  [13, Corollary 2.100].) We prove that the continuity of f implies that of f^ if X is locally compact and Y is regular, as it is apparently not so standard.
Suppose thatf is continuous and f^(z0,x0)=:y0 lies in the open set V⊂Y.  As Y is regular, there is an open set W with y0∈W⊂W¯⊂V.  As X is locally compact, x0 is contained in the interior of a compact set C. Write ϕ=f(z0). Clearly, K=ϕ-1(W¯)∩C  is a compact set contained in ϕ-1(V). This means that f(z0) lies in the open set G(K,V). As f is continuous, there is an open neighbourhood U of z0 so that f(U)⊂G(K,V). Consequently, f^(U×K)⊂V. As x0 lies in the interior of K, f^ is continuous at (z0,x0).Definition 4.
For any spaceZ, let κ(Z×X) denote the topological space whose underlying set is Z×X and has its topology determined by the subsets Z×C (with the Cartesian product topology) where C ranges over the compact subsets of X. That is, F⊂Z×X is closed in κ(Z×X) if and only if F∩(Z×C) is closed in Z×C for each compact subspace C of X. The identity κ(Z×X)→Z×X, where the latter has the Cartesian product topology, is evidently continuous.In the language of Dydak [14], κ(Z×X) has the covariant topology on Z×X induced by the class of set-theoretic inclusions Z×C↪Z×X where C ranges over the compact subsets of X and the Z×C carry the product topology.The introduction of the topologyκ(Z×X) is motivated by the following lemma.Lemma 5.
LetZ be any topological space, let Y be a regular space, and let X be a compactly generated hemicompact space. Let f:Z→YX be a function with set-theoretic adjoint f^:X×Z→Y. Then f is continuous if and only if f^:κ(Z×X)→Y is continuous. This accounts for a bijection (YX)Z↔Yκ(Z×X).Proof.
Let{Ki} be the sequence of distinguished compacta in X and let Ri:YX→YKi denote the map that to each function X→Y assigns its restriction to Ki. Clearly Ri is continuous. If f:Z→YX is continuous, then so is the composite Ri∘f:Z→YKi. By  Proposition 3, so is its adjoint f^|Z×Ki:Z×Ki→Y. As each compact set C is contained in one of the Ki, it follows by definition of κ(Z×X) that f^:κ(Z×X)→Y is continuous.
For the converse, assume thatf^:κ(Z×X)→Y is continuous. This means that the restrictions of f^ to subspaces Z×Ki are continuous, and  Proposition 3 implies that the composites Ri∘f:Z→YKi are continuous. But as X is compactly generated, a function ϕ:X→Y is continuous if and only if all restrictions ϕ|Ki:Ki→Y are continuous. This means that the obvious map Z→∏i=1∞‍YKi, whose components are Ri∘f, maps into the image of the embedding (*) and therefore yields a continuous function Z→YX which is precisely f.Lemma 6.
LetK be a compact regular space. The topologies κ((Z×K)×X) and κ(Z×X)×K (viewed as topologies on Z×X×K) coincide.We note that this is a corollary of the much more general Theorem 1.15 of Dydak [14]. (Since K is compact regular, it is locally compact according to the definition in [14].) For the sake of completeness, we provide an independent proof (along slightly different lines).Proof.
We show that the two topologies have the same continuous maps into an arbitrary spaceY. By definition, f:κ(Z×X×K)→Y is continuous if and only if the restrictions fC:Z×C×K→Y (for compact C⊂X) are continuous which, by  Proposition 3, is if and only if their adjoints f^C:Z×C→YK are continuous. The latter is if and only if the map f^:κ(Z×X)→YK is continuous and this in turn if and only if f:κ(Z×X)×K→Y is continuous, by another application of  Proposition 3. This finishes the proof.Let(Z,A) be a topological pair (no separation properties assumed). Then A is P-embedded in Z if continuous pseudometrics on A extend to continuous pseudometrics on Z. Also, A is a zero set in Z if there exists a continuous function ϕ:Z→ℝ with A=ϕ-1(0). If A is a P-embedded zero set, it is called P0-embedded.For example, every closed subset of a metrizable space isP0-embedded.We needP-embeddings in the context of Morita’s homotopy extension theorem (which in fact characterizes ANR spaces; see Stramaccia [15]).Theorem 7 (Morita [8]).
IfA is P0-embedded in the topological space Z, then the pair (Z,A) has the homotopy extension property with respect to all ANR spaces. That is, if Y is an ANR, if g:Z×{0}→Y and h:A×[0,1]→Y are continuous maps that agree pointwise on A×{0}, then there exists a continuous map H:Z×[0,1]→Y extending both g and h.Lemma 8.
LetE be a Fréchet space and let X be a compactly generated hemicompact (not necessarily Hausdorff) space. Then EX is also a Fréchet space.Proof.
One verifies readily thatEX is a topological vector space and that the subbasic open sets G(K,V), where V are convex neighbourhoods of 0 in E, constitute a convex local base for EX (see Schaefer [16], page 80). If E is metrizable (by an invariant metric), then EX is metrizable by the (invariant) metric (**) above. If E is complete, then so is EX since X is compactly generated (see, e.g., Willard [6, Theorem 43.11]).Proposition 9.
LetA be P-embedded in Z and let X be a compactly generated hemicompact space. Then the subset A×X is P-embedded in κ(Z×X).A result due to Alò and Sennott (see [17, Theorem 1.2]) shows that A is P-embedded in Z if and only if every continuous function from A to a Fréchet space extends continuously over Z.  Proposition 9 seems to be the right way of generalizing the equivalence (1)  ⇔  (2) of Theorem 2.4 in [17].For a closed subsetA of Z, the topology κ(A×X) coincides with the topology that the set A×X inherits from κ(Z×X). For arbitrary A, the two topologies may differ, but note that κ(A×X) is always finer than the subspace topology.Proof.
LetE be a Fréchet space and let f:A×X→E be a continuous map where A×X is understood to inherit its topology from κ(Z×X). Precomposing with the continuous identity κ(A×X)→A×X and using  Lemma 5, we obtain a continuous map A→EX. By  Lemma 8, EX is also a Fréchet space and as A is P-embedded in Z, the function f^ extends continuously to F^:Z→EX. Reapplying  Lemma 5, F^ induces the desired extension F:κ(Z×X)→E.Proof of Theorem1.
LetZ be metrizable and let f:A→YX be a continuous map defined on the closed subset A of Z. By assumption, YX has the homotopy type of an ANR; hence f admits a neighbourhood extension up to homotopy. That is, there exist a continuous map g:U→YX where U is open and contains A and a homotopy h:A×[0,1]→YX beginning in g|A and ending in f.
Leth⊔g:A×[0,1]∪U×{0}→YX denote the continuous union of the two, with adjoint h^⊔g^:κ((A×[0,1]∪U×{0})×X)→Y. As A×[0,1]∪U×{0} is closed in A×U, the map h^⊔g^ is continuous with respect to the topology that (A×[0,1]∪U×{0})×X inherits from κ(U×[0,1]×X). Under the homeomorphism κ(U×[0,1]×X)≈κ(U×X)×[0,1] of  Lemma 6, the map h^⊔g^ corresponds to a continuous map k^:κ(A×X)×[0,1]∪κ(U×X)×{0}→Y.
Obviously, asA is a zero set in U, the product A×X is a zero set in U×X with respect to the Cartesian product topology. A fortiori, A×X is a zero set in κ(U×X). Hence, by  Proposition 9, the set A×X is P0-embedded in κ(U×X).  Theorem 7 yields an extension of k^ to K:κ(U×X)×[0,1]→Y. Reapplying  Lemma 6 and  Lemma 5, K induces a continuous function k:U×[0,1]→YX. Level 1 of this homotopy is a continuous extension of f over the neighbourhood U. Therefore, YX is an ANR.Corollary 10.
IfC is a compact space and Y is an ANR, then YC is an ANR.Proof.
By Theorem 3 of Milnor [3], YC has CW homotopy type.Corollary10 was proved independently by Yamashita [18] (with the additional requirement that C be Hausdorff) but the author of this note has not seen it elsewhere for nonmetrizable compacta C. From the point of view of P-embeddings, however,  Corollary 10 encodes a long-known fact (see Przymusiński [19, Theorem 3]): if A is P-embedded in Z and X is a compact space, then A×X is P-embedded in Z×X.Proof of Corollary2.
Suppose thatYX is metrizable. If Y contains an arc (which is if and only if it has a nontrivial path component), it follows that [0,1]X is metrizable. Since X is a Tychonoff space, points in X can be separated from compact sets in X by means of continuous functions X→[0,1]. The proof of Theorem 8 of Arens [5] can be adapted almost verbatim to render X hemicompact. The statement of  Corollary 2 follows immediately from  Theorem 1.



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