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Homotopy Characterization of ANR Function Spaces Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 925742, 5 pages http://dx.doi.org/10.1155/2013/925742 Research Article Homotopy Characterization of ANR Function Spaces Jaka Smrekar Fakulteta za Matematiko in Fiziko, Jadranska Ulica 19, 1111 Ljubljana, Slovenia Correspondence should be addressed to Jaka Smrekar; [email protected] Received 5 May 2013; Accepted 25 August 2013 Academic Editor: Yongsheng S. Han Copyright © 2013 Jaka Smrekar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 typeofaCWcomplex. 1. Introduction over the convergent sequence {0,1,1/2,1/3,...}.Then is contractible; that is, it has the homotopy type of a one-point Let be an absolute neighbourhood retract for metric spaces CW complex. But as is not locally path connected, it is not (henceforth abbreviated as “ANR”). This means that when- an ANR. ever is embedded in a metric space as a closed subspace On the other hand, Cauty [4]showedthatametrizable ̂, there exists a retraction of an open neighbourhood onto space is an ANR if and only if each open subspace has ̂. We refer the reader to the first part of Mardeˇsic[´ 1]fora the homotopy type of a CW complex. It turns out that the scenic survey of the theory of ANRs. space of functions into an ANR inherits a good deal of reasonable behaviour from the target space. Thus, under mild Let be a topological space. The question that this paper restrictions on ,if is an ANR and is a metrizable space is concerned with is when is , the space of continuous → with the homotopy type of a CW complex, is in fact an functions , equipped with the compact open ANR. Put another way, is an ANR if and only if is an topology, also an ANR. A basic result of Kuratowski (see [2, page 284]), which is a consequence of the classical homotopy ANR and is a metrizable space with the homotopy type of extension theorem of Borsuk, states that is an ANR if is an ANR. ametrizablecompactum. Basic Definitions and Conventions. A topological space is For a negative example, consider the discrete space of N {0, 1} called hemicompact if is the union of countably many natural numbers and the two-point discrete ANR . { |} {0, 1}N of its compact subsets which dominate all compact Then is a Cantor set and hence certainly not an ANR. subsets in . This means that for each compact ⊂there {0, 1}N In fact, as the path components of are not open, it does exists with ⊂. (The perhaps somewhat noninformative not even have the homotopy type of a CW complex; that is, word “hemicompact” was introduced by Arens [5]inrelation it is not homotopy equivalent to any CW complex. As every to metrizability of function spaces. See the beginning of ANR has the homotopy type of a CW complex, this provides Section 2.) anecessaryconditionfor to be an ANR. Aspace is compactly generated if the compact sub- In fact, a topological space has the homotopy type of spaces determine its topology. That is, a subset is closed an ANR if and only if it has the homotopy type of a CW in if and only if ∩is closed in for each compact complex(seeMilnor[3,Theorem2]).However,thereare subspace . Such spaces are also commonly called -spaces numerousexamplesofspacesthathaveCWhomotopytype (see, e.g., Willard [6]). We do not require a hemicompact or but are not ANRs. For example, let be the topological cone a compactly generated space to be Hausdorff. 2 Journal of Function Spaces and Applications Aspace is Tychonoff if it is both completely regular and continuous function →,where is a closed subspace of Hausdorff. Locally compact Hausdorff spaces and normal a metric space, extends continuously over a neighbourhood Hausdorff spaces are Tychonoff (examples of the latter are all of . metric spaces and all CW complexes). Note that if is a hemicompact space with the sequence The terms map and continuous function will be used of “distinguished” compact sets {}, the map into the synonymously. countable Cartesian product Amap: → is a homotopy equivalence if there : → ∞ exists a map (called a homotopy inverse) → ∏ , defined by → { | } (∗) for which the composites ∘and ∘are homotopic to =1 their respective identities. In this case, and are called homotopy equivalent,andwesaythat has the homotopy is an embedding (see also Cauty [12]). Consequently, if type of . denotes the supmetric on induced by a metric on ,then The following are our main results. is metrizable by the metric Theorem 1. Let be a compactly generated hemicompact ∞ 1 space and let be an ANR. Then is an ANR if and only (,)=∑ min { , (| ,| )} . (∗∗) 2 if has the homotopy type of an ANR, which is if and only if =1 it has the homotopy type of a CW complex. (See Arens [5,Theorem7].)Giventhehypothesesof We call a (not necessarily Hausdorff) space locally com- Theorem 1, therefore, we need to show that for every pair pact if each point is contained in the interior of a compact (, ) with metric and closed in ,everycontinuous set. It is well-known that compactly generated spaces are pre- function : → extends continuously over a cisely quotient spaces of locally compact spaces. Compactly neighbourhood of in . We need some preliminary results. generated hemicompact spaces seem to be important enough First, we state the classical exponential correspondence to warrant an analogous characterization. In the appendix, theorem with minimal hypotheses. Here, a space is regular we prove that they arise as nice quotient spaces of -compact if points can be separated from closed sets by disjoint open locally compact spaces. sets. Assuming additional separation properties, Theorem 1 canbestrengthenedasfollows. Proposition 3. Let , ,and be topological spaces. Let : → be any function with set-theoretic adjoint Corollary 2. Let be a compactly generated Tychonoff space :×→̂ ̂ .If is continuous, then is (well-defined and let be an ANR which contains an arc. Then is an ANR and) continuous. For the converse, suppose that is locally if and only if is hemicompact and has the homotopy type compact. If is continuous and, in addition, is regular or ̂ of a CW complex. is regular, then is continuous. This accounts for a bijection () ↔(×) Theorem 1 is a considerable extension of Theorem 1.1 . of [7] where the equivalence was proved using a different Proof. The requirement that be regular is standard. (See, technique under the more stringent requirement that be e.g., [13, Corollary 2.100].) We prove that the continuity of a countable CW complex. Our proof of Theorem 1 leans ̂ implies that of if is locally compact and is regular, as it on Morita’s homotopy extension theorem for 0-embeddings (see Morita [8]). is apparently not so standard. (̂ , )=: Even when is a countable CW complex, it is highly Suppose that is continuous and 0 0 0 lies in nontrivial to determine whether or not the function space theopenset⊂.As is regular, there is an open set has the homotopy type of a CW complex. The interested with 0 ∈⊂⊂.As is locally compact, 0 is reader is referred to papers [7, 9, 10] for more on this. contained in the interior of a compact set .Write=(0). −1 −1 Clearly, = ()∩ is a compact set contained in (). This means that (0) lies in the open set (,.As ) is continuous, there is an open neighbourhood of 0 so that 2. Proof of Theorem 1 ̂ () ⊂ (,.Consequently, ) ( × ).As ⊂ 0 lies in For subsets of the domain space and of the target space, ̂ the interior of , is continuous at (0,0). we let (, ) denote the set of all maps that map the set into the set . For topological spaces and ,thestandard ( × ) Definition 4. For any space ,let denote the subbasis of the compact open topology on is the collection topological space whose underlying set is ×and has its P of all (, ) ⊂ with a compact subset of and topology determined by the subsets ×(with the Cartesian an open subset of . product topology) where ranges over the compact subsets To prove Theorem 1, we use the fact that ANRs for metric of .Thatis,⊂×is closed in ( × ) if and only if spacesarepreciselythemetrizableabsoluteneighbourhood ∩ ( ×) is closed in ×for each compact subspace of extensors for metric spaces (abbreviated as “ANE”); see, for . The identity ( × ),wherethelatterhasthe →× example, Hu [11, Theorem 3.2]. A space is an ANE if every Cartesian product topology, is evidently continuous. Journal of Function Spaces and Applications 3 −1 In the language of Dydak [14], ( × ) has the covariant : → R with = (0).If is a -embedded zero set, topology on ×induced by the class of set-theoretic it is called 0-embedded. inclusions ×→×where ranges over the compact For example, every closed subset of a metrizable space is subsets of and the ×carry the product topology. 0-embedded. The introduction of the topology ( × ) is motivated We need -embeddings in the context of Morita’s homo- by the following lemma. topy extension theorem (which in fact characterizes ANR spaces; see Stramaccia [15]). Lemma 5. Let be any topological space, let be a regular space, and let be a compactly generated hemicompact space. Theorem 7 (Morita [8]).
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