Homotopy Characterization of ANR Function Spaces
Let be an absolute neighbourhood retract (ANR) for the class of metric spaces and let be a topological space. Let denote the space of continuous maps from to equipped with the compact open topology. We show that if is a compactly generated Tychonoff space and is not discrete, then is an ANR for metric spaces if and only if is hemicompact and has the homotopy type of a CW complex.
Let be an absolute neighbourhood retract for metric spaces (henceforth abbreviated as “ANR”). This means that whenever is embedded in a metric space as a closed subspace , there exists a retraction of an open neighbourhood onto . We refer the reader to the first part of Mardešić  for a scenic survey of the theory of ANRs.
Let be a topological space. The question that this paper is concerned with is when is , the space of continuous functions , 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 is an ANR if is a metrizable compactum.
For a negative example, consider the discrete space of natural numbers and the two-point discrete ANR . Then is a Cantor set and hence certainly not an ANR. In fact, as the path components of 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 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 be the topological cone over the convergent sequence . Then is contractible; that is, it has the homotopy type of a one-point CW complex. But as is not locally path connected, it is not an ANR.
On the other hand, Cauty  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 , if is an ANR and is a metrizable space with the homotopy type of a CW complex, is in fact an ANR. Put another way, is an ANR if and only if is an ANR and is a metrizable space with the homotopy type of an ANR.
Basic Definitions and Conventions. A topological space is called hemicompact if is the union of countably many of its compact subsets which dominate all compact subsets in . This means that for each compact there exists with . (The perhaps somewhat noninformative word “hemicompact” was introduced by Arens  in relation to metrizability of function spaces. See the beginning of Section 2.)
A space is compactly generated if the compact subspaces determine its topology. That is, a subset is closed in if and only if is closed in for each compact subspace . Such spaces are also commonly called spaces (see, e.g., Willard ). We do not require a hemicompact or a compactly generated space to be Hausdorff.
A space 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 terms map and continuous function will be used synonymously.
A map is a homotopy equivalence if there exists a map (called a homotopy inverse) for which the composites and are homotopic to their respective identities. In this case, and are called homotopy equivalent, and we say that has the homotopy type of .
The following are our main results.
Theorem 1. Let be a compactly generated hemicompact space and let be an ANR. Then is an ANR if and only if 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 of -compact locally compact spaces.
Assuming additional separation properties, Theorem 1 can be strengthened as follows.
Corollary 2. Let be a compactly generated Tychonoff space and let be an ANR which contains an arc. Then is an ANR if and only if is hemicompact and has the homotopy type of a CW complex.
Theorem 1 is a considerable extension of Theorem 1.1 of  where the equivalence was proved using a different technique under the more stringent requirement that be a countable CW complex. Our proof of Theorem 1 leans on Morita’s homotopy extension theorem for -embeddings (see Morita ).
Even when is a countable CW complex, it is highly nontrivial to determine whether or not the function space 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 Theorem 1
For subsets of the domain space and of the target space, we let denote the set of all maps that map the set into the set . For topological spaces and , the standard subbasis of the compact open topology on is the collection of all with a compact subset of and an open subset of .
To prove Theorem 1, 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 is an ANE if every continuous function , where is a closed subspace of a metric space, extends continuously over a neighbourhood of .
Note that if is a hemicompact space with the sequence of “distinguished” compact sets , the map into the countable Cartesian product is an embedding (see also Cauty ). Consequently, if denotes the metric on induced by a metric on , then is metrizable by the metric (See Arens [5, Theorem 7].) Given the hypotheses of Theorem 1, therefore, we need to show that for every pair with metric and closed in , every continuous function extends continuously over a neighbourhood of in . 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. Let , , and be topological spaces. Let be any function with set-theoretic adjoint . If is continuous, then is (well-defined and) continuous. For the converse, suppose that is locally compact. If is continuous and, in addition, is regular or is regular, then is continuous. This accounts for a bijection .
Proof. The requirement that be regular is standard. (See, e.g., [13, Corollary 2.100].) We prove that the continuity of implies that of if is locally compact and is regular, as it is apparently not so standard.
Suppose that is continuous and lies in the open set . As is regular, there is an open set with . As is locally compact, is contained in the interior of a compact set . Write . Clearly, is a compact set contained in . This means that lies in the open set . As is continuous, there is an open neighbourhood of so that . Consequently, . As lies in the interior of , is continuous at .
Definition 4. For any space , let denote the topological space whose underlying set is and has its topology determined by the subsets (with the Cartesian product topology) where ranges over the compact subsets of . That is, is closed in if and only if is closed in for each compact subspace of . The identity , where the latter has the Cartesian product topology, is evidently continuous.
In the language of Dydak , has the covariant topology on induced by the class of set-theoretic inclusions where ranges over the compact subsets of and the carry the product topology.
The introduction of the topology is motivated by the following lemma.
Lemma 5. Let be any topological space, let be a regular space, and let be a compactly generated hemicompact space. Let be a function with set-theoretic adjoint . Then is continuous if and only if is continuous. This accounts for a bijection .
Proof. Let be the sequence of distinguished compacta in and let denote the map that to each function assigns its restriction to . Clearly is continuous. If is continuous, then so is the composite . By Proposition 3, so is its adjoint . As each compact set is contained in one of the , it follows by definition of that is continuous.
For the converse, assume that is continuous. This means that the restrictions of to subspaces are continuous, and Proposition 3 implies that the composites are continuous. But as is compactly generated, a function is continuous if and only if all restrictions are continuous. This means that the obvious map , whose components are , maps into the image of the embedding and therefore yields a continuous function which is precisely .
Lemma 6. Let be a compact regular space. The topologies and (viewed as topologies on ) coincide.
We note that this is a corollary of the much more general Theorem 1.15 of Dydak . (Since is compact regular, it is locally compact according to the definition in .) 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 space . By definition, is continuous if and only if the restrictions (for compact ) are continuous which, by Proposition 3, is if and only if their adjoints are continuous. The latter is if and only if the map is continuous and this in turn if and only if is continuous, by another application of Proposition 3. This finishes the proof.
Let be a topological pair (no separation properties assumed). Then is -embedded in if continuous pseudometrics on extend to continuous pseudometrics on . Also, is a zero set in if there exists a continuous function with . If is a -embedded zero set, it is called -embedded.
For example, every closed subset of a metrizable space is -embedded.
We need -embeddings in the context of Morita’s homotopy extension theorem (which in fact characterizes ANR spaces; see Stramaccia ).
Theorem 7 (Morita ). If is -embedded in the topological space , then the pair has the homotopy extension property with respect to all ANR spaces. That is, if is an ANR, if and are continuous maps that agree pointwise on , then there exists a continuous map extending both and .
Lemma 8. Let be a Fréchet space and let be a compactly generated hemicompact (not necessarily Hausdorff) space. Then is also a Fréchet space.
Proof. One verifies readily that is a topological vector space and that the subbasic open sets , where are convex neighbourhoods of in , constitute a convex local base for (see Schaefer , page 80). If is metrizable (by an invariant metric), then is metrizable by the (invariant) metric above. If is complete, then so is since is compactly generated (see, e.g., Willard [6, Theorem 43.11]).
Proposition 9. Let be -embedded in and let be a compactly generated hemicompact space. Then the subset is -embedded in .
A result due to Alò and Sennott (see [17, Theorem 1.2]) shows that is -embedded in if and only if every continuous function from to a Fréchet space extends continuously over . Proposition 9 seems to be the right way of generalizing the equivalence (1) (2) of Theorem 2.4 in .
For a closed subset of , the topology coincides with the topology that the set inherits from . For arbitrary , the two topologies may differ, but note that is always finer than the subspace topology.
Proof. Let be a Fréchet space and let be a continuous map where is understood to inherit its topology from . Precomposing with the continuous identity and using Lemma 5, we obtain a continuous map . By Lemma 8, is also a Fréchet space and as is -embedded in , the function extends continuously to . Reapplying Lemma 5, induces the desired extension .
Proof of Theorem 1. Let be metrizable and let be a continuous map defined on the closed subset of . By assumption, has the homotopy type of an ANR; hence admits a neighbourhood extension up to homotopy. That is, there exist a continuous map where is open and contains and a homotopy beginning in and ending in .
Let denote the continuous union of the two, with adjoint . As is closed in , the map is continuous with respect to the topology that inherits from . Under the homeomorphism of Lemma 6, the map corresponds to a continuous map .
Obviously, as is a zero set in , the product is a zero set in with respect to the Cartesian product topology. A fortiori, is a zero set in . Hence, by Proposition 9, the set is -embedded in . Theorem 7 yields an extension of to . Reapplying Lemma 6 and Lemma 5, induces a continuous function . Level of this homotopy is a continuous extension of over the neighbourhood . Therefore, is an ANR.
Corollary 10. If is a compact space and is an ANR, then is an ANR.
Proof. By Theorem 3 of Milnor , has CW homotopy type.
Corollary 10 was proved independently by Yamashita  (with the additional requirement that be Hausdorff) but the author of this note has not seen it elsewhere for nonmetrizable compacta . From the point of view of -embeddings, however, Corollary 10 encodes a long-known fact (see Przymusiński [19, Theorem 3]): if is -embedded in and is a compact space, then is -embedded in .
Proof of Corollary 2. Suppose that is metrizable. If contains an arc (which is if and only if it has a nontrivial path component), it follows that is metrizable. Since is a Tychonoff space, points in can be separated from compact sets in by means of continuous functions . The proof of Theorem 8 of Arens  can be adapted almost verbatim to render hemicompact. The statement of Corollary 2 follows immediately from Theorem 1.
A Characterization of Compactly Generated Hemicompact Spaces
A function will be called weakly proper if for each compact subset of there exists a compact subset of so that . Note that a weakly proper map is necessarily surjective. Finally, recall that is a -compact space if is the union of a countable collection of its compact subsets.
Proposition A.1. The topological space is compactly generated and hemicompact if and only if there exists a -compact locally compact space with a weakly proper quotient map .
Proof. Let be a compactly generated hemicompact space with its sequence of distinguished compact subsets and let denote the obvious surjective map from the disjoint union to . Clearly, a set in is closed if and only if is closed in for all . Thus, is a quotient map. Tautologically, is both -compact and locally compact, and hemicompactness renders weakly proper.
For the reverse implication, suppose first that is a compactly generated hemicompact space with distinguished compact subsets and is a weakly proper quotient map. Clearly, as is weakly proper, is hemicompact with distinguished compact subsets . To see that is compactly generated, let be a function that is continuous on all compact sets. Hence, if is a compact subset of , is continuous on the saturation . Consequently, is continuous on . Since is compactly generated, is continuous and since is a quotient map, so also is . Since this holds for all , is compactly generated.
Finally, suppose that is a -compact locally compact space. Then is compactly generated. (See, e.g., , 43.9.) Since is locally compact, it has an open cover consisting of relatively compact sets. As is the union of countably many compact sets, has a countable subcover . The sequence of compact sets , , exhibits as hemicompact. This completes the proof.
The author is grateful to Atsushi Yamashita for posing the question that has led to the results presented here and to the referee for recommending that a characterization of compactly generated hemicompact spaces in the sense of Proposition A.1 be included in the paper. The author was supported in part by the Slovenian Research Agency Grants P1-0292-0101 and J1-4144-0101.
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