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.

𝑋 Let 𝑌 be an absolute neighbourhood retract (ANR) for the class of metric spaces and let 𝑋 be a topological space. Let 𝑌 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 𝑋 𝑋 Tychonoff space and 𝑌 is not discrete, then 𝑌 is an ANR for metric spaces if and only if 𝑋 is hemicompact and 𝑌 has the homotopy typeofaCWcomplex.

1. Introduction over the convergent sequence {0,1,1/2,1/3,...}.Then𝐹 is contractible; that is, it has the homotopy type of a one-point Let 𝑌 be an absolute neighbourhood retract for metric spaces CW complex. But as 𝐹 is not locally path connected, it is not (henceforth abbreviated as “ANR”). This means that when- an ANR. ever 𝑌 is embedded in a metric space as a closed subspace On the other hand, Cauty [4]showedthatametrizable 𝑌̂, there exists a retraction of an open neighbourhood onto space is an ANR if and only if each open subspace has 𝑌̂. 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 𝑋 reasonable behaviour from the target space. Thus, under mild Let be a topological space. The question that this paper 𝑋 𝑌𝑋 restrictions on 𝑋,if𝑌 is an ANR and 𝑌 is a metrizable space is concerned with is when is , the space of continuous 𝑋 𝑋→𝑌 with the homotopy type of a CW complex, 𝑌 is in fact an functions , equipped with the compact open 𝑋 ANR. Put another way, 𝑌 is an ANR if and only if 𝑌 is an topology, also an ANR. A basic result of Kuratowski (see [2, 𝑋 page 284]), which is a consequence of the classical homotopy ANR and 𝑌 is a metrizable space with the homotopy type of 𝑋 extension theorem of Borsuk, states that 𝑌 is an ANR if 𝑋 is an ANR. ametrizablecompactum. Basic Definitions and Conventions. A topological space 𝑋 is For a negative example, consider the discrete space of 𝑋 N {0, 1} called hemicompact if is the union of countably many natural numbers and the two-point discrete ANR . {𝐾 |𝑖} {0, 1}N of its compact subsets 𝑖 which dominate all compact Then is a Cantor set and hence certainly not an ANR. subsets in 𝑋. This means that for each compact 𝐶⊂𝑋there {0, 1}N In fact, as the path components of are not open, it does exists 𝑖 with 𝐶⊂𝐾𝑖. (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 𝑋 Section 2.) anecessaryconditionfor𝑌 to be an ANR. Aspace𝑋 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 𝐴 is closed an ANR if and only if it has the homotopy type of a CW in 𝑋 if and only if 𝐴∩𝐶is closed in 𝐶 for each compact complex(seeMilnor[3,Theorem2]).However,thereare subspace 𝐶. Such spaces are also commonly called 𝑘-spaces numerousexamplesofspacesthathaveCWhomotopytype (see, e.g., Willard [6]). We do not require a hemicompact or but are not ANRs. For example, let 𝐹 be the topological cone a compactly generated space to be Hausdorff. 2 Journal of Function Spaces and Applications

Aspace𝑋 is Tychonoff if it is both completely regular and continuous function 𝐴→𝑌,where𝐴 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 𝐴. metric spaces and all CW complexes). Note that if 𝑋 is a hemicompact space with the sequence The terms map and continuous function will be used of “distinguished” compact sets {𝐾𝑖}, the map into the synonymously. countable Cartesian product 󸀠 Amap𝑓:𝑌 →𝑌 is a homotopy equivalence if there 𝑔:𝑌󸀠 →𝑌 ∞ exists a map (called a homotopy inverse) 𝑋 𝐾𝑖 𝑌 󳨀→ ∏𝑌 , defined by 𝑓 󳨃󳨀→ { 𝑓| 𝐾 } (∗) for which the composites 𝑔∘𝑓and 𝑓∘𝑔are homotopic to 𝑖 󸀠 𝑖=1 their respective identities. In this case, 𝑌 and 𝑌 are called 󸀠 homotopy equivalent,andwesaythat𝑌 has the homotopy is an embedding (see also Cauty [12]). Consequently, if 𝑑𝑖 type of 𝑌. 𝐾𝑖 denotes the supmetric on 𝑌 induced by a metric on 𝑌,then The following are our main results. 𝑋 𝑌 is metrizable by the metric Theorem 1. 𝑋 Let be a compactly generated hemicompact ∞ 𝑌 𝑌𝑋 1 space and let be an ANR. Then is an ANR if and only 𝑑(𝑓,𝑔)=∑ min { ,𝑑𝑖 (𝑓|𝐾 ,𝑔|𝐾 )} . (∗∗) 𝑋 2𝑖 𝑖 𝑖 if 𝑌 has the homotopy type of an ANR, which is if and only if 𝑖=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 (𝑍, 𝐴) with 𝑍 metric and 𝐴 closed in 𝑍,everycontinuous 𝑋 set. It is well-known that compactly generated spaces are pre- function 𝑓:𝐴 →𝑌 extends continuously over a cisely quotient spaces of locally compact spaces. Compactly neighbourhood of 𝐴 in 𝑍. 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 𝜎-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 𝑋, 𝑌,and𝑍 be topological spaces. Let 𝑋 𝑓:𝑍 →𝑌 be any function with set-theoretic adjoint Corollary 2. Let 𝑋 be a compactly generated Tychonoff space 𝑓:𝑍×𝑋→̂ 𝑌 𝑓̂ 𝑓 𝑋 .If is continuous, then is (well-defined and let 𝑌 be an ANR which contains an arc. Then 𝑌 is an ANR 𝑋 𝑋 and) continuous. For the converse, suppose that is locally if and only if 𝑋 is hemicompact and 𝑌 has the homotopy type compact. If 𝑓 is continuous and, in addition, 𝑋 is regular or ̂ of a CW complex. 𝑌 is regular, then 𝑓 is continuous. This accounts for a bijection (𝑌𝑋)𝑍 ↔𝑌(𝑋×𝑍) Theorem 1 is a considerable extension of Theorem 1.1 . of [7] where the equivalence was proved using a different Proof. The requirement that 𝑋 be regular is standard. (See, technique under the more stringent requirement that 𝑋 be e.g., [13, Corollary 2.100].) We prove that the continuity of 𝑓 a countable CW complex. Our proof of Theorem 1 leans ̂ implies that of 𝑓 if 𝑋 is locally compact and 𝑌 is regular, as it on Morita’s homotopy extension theorem for 𝑃0-embeddings (see Morita [8]). is apparently not so standard. 𝑓 𝑓(𝑧̂ ,𝑥 )=:𝑦 Even when 𝑋 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 theopenset𝑉⊂𝑌.As𝑌 is regular, there is an open set 𝑋 𝑌 has the homotopy type of a CW complex. The interested 𝑊 with 𝑦0 ∈𝑊⊂𝑊⊂𝑉.As𝑋 is locally compact, 𝑥0 is reader is referred to papers [7, 9, 10] for more on this. contained in the interior of a compact set 𝐶.Write𝜙=𝑓(𝑧0). −1 −1 Clearly, 𝐾=𝜙 (𝑊)∩𝐶 is a compact set contained in 𝜙 (𝑉). This means that 𝑓(𝑧0) lies in the open set 𝐺(𝐾,.As 𝑉) 𝑓 is continuous, there is an open neighbourhood 𝑈 of 𝑧0 so that 2. Proof of Theorem 1 ̂ 𝑓(𝑈) ⊂ 𝐺(𝐾,.Consequently, 𝑉) 𝑓(𝑈 × 𝐾) ⊂𝑉.As𝑥0 lies in For subsets 𝐴 of the domain space and 𝐵 of the target space, ̂ the interior of 𝐾, 𝑓 is continuous at (𝑧0,𝑥0). 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,thestandard 𝑍 𝜅(𝑍 × 𝑋) 𝑋 Definition 4. For any space ,let denote the subbasis of the compact open topology on 𝑌 is the collection topological space whose underlying set is 𝑍×𝑋and has its 𝑋 P of all 𝐺(𝐾, 𝑉) ⊂𝑌 with 𝐾 a compact subset of 𝑋 and 𝑉 topology determined by the subsets 𝑍×𝐶(with the Cartesian an open subset of 𝑌. product topology) where 𝐶 ranges over the compact subsets To prove Theorem 1, we use the fact that ANRs for metric of 𝑋.Thatis,𝐹⊂𝑍×𝑋is closed in 𝜅(𝑍 × 𝑋) if and only if spacesarepreciselythemetrizableabsoluteneighbourhood 𝐹 ∩ (𝑍 ×𝐶) is closed in 𝑍×𝐶for each compact subspace 𝐶 of extensors for metric spaces (abbreviated as “ANE”); see, for 𝑋. The identity 𝜅(𝑍 × 𝑋),wherethelatterhasthe →𝑍×𝑋 example, Hu [11, Theorem 3.2]. A space 𝑌 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], 𝜅(𝑍 × 𝑋) has the covariant 𝜙:𝑍 → R with 𝐴=𝜙 (0).If𝐴 is a 𝑃-embedded zero set, topology on 𝑍×𝑋induced by the class of set-theoretic it is called 𝑃0-embedded. inclusions 𝑍×𝐶󳨅→𝑍×𝑋where 𝐶 ranges over the compact For example, every closed subset of a metrizable space is subsets of 𝑋 and the 𝑍×𝐶carry the product topology. 𝑃0-embedded. The introduction of the topology 𝜅(𝑍 × 𝑋) is motivated We need 𝑃-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 𝑍 be any topological space, let 𝑌 be a regular space, and let 𝑋 be a compactly generated hemicompact space. Theorem 7 (Morita [8]). If 𝐴 is 𝑃0-embedded in the topolog- 𝑋 ̂ Let 𝑓:𝑍 →𝑌 be a function with set-theoretic adjoint 𝑓: ical space 𝑍, then the pair (𝑍, 𝐴) has the homotopy extension ̂ 𝑌 𝑋×𝑍→𝑌.Then𝑓 is continuous if and only if 𝑓:𝜅(𝑍× property with respect to all ANR spaces. That is, if is an ANR, 𝑔:𝑍×{0}→𝑌 ℎ:𝐴×[0,1]→ 𝑌 𝑋) → 𝑌 (𝑌𝑋)𝑍 ↔ if and are continuous maps is continuous. This accounts for a bijection 𝐴×{0} 𝑌𝜅(𝑍×𝑋) that agree pointwise on , then there exists a continuous . map 𝐻:𝑍×[0,1]→extending 𝑌 both 𝑔 and ℎ.

Proof. Let {𝐾𝑖} be the sequence of distinguished compacta 𝑋 𝐾 Lemma 8. Let 𝐸 be a Frechet´ space and let 𝑋 be a compactly 𝑋 𝑅 :𝑌 →𝑌𝑖 in and let 𝑖 denote the map that to each generated hemicompact (not necessarily Hausdorff) space. function 𝑋→𝑌assigns its restriction to 𝐾𝑖.Clearly𝑅𝑖 is 𝑋 𝑋 Then 𝐸 is also a Frechet´ space. continuous. If 𝑓:𝑍 →𝑌 is continuous, then so is the 𝐾 𝑅 ∘𝑓:𝑍→𝑌𝑖 𝑋 composite 𝑖 .ByProposition 3,soisits Proof. One verifies readily that 𝐸 is a topological vector 𝑓|̂ :𝑍×𝐾 →𝑌 𝐶 𝐺(𝐾, 𝑉) 𝑉 adjoint 𝑍×𝐾𝑖 𝑖 . As each compact set is space and that the subbasic open sets ,where are contained in one of the 𝐾𝑖, it follows by definition of 𝜅(𝑍×𝑋) convex neighbourhoods of 0 in 𝐸,constituteaconvexlocal ̂ 𝑋 that 𝑓:𝜅(𝑍×𝑋)→is 𝑌 continuous. base for 𝐸 (see Schaefer [16], page 80). If 𝐸 is metrizable (by ̂ 𝑋 For the converse, assume that 𝑓:𝜅(𝑍×𝑋)→is 𝑌 an invariant metric), then 𝐸 is metrizable by the (invariant) ̂ 𝑋 continuous. This means that the restrictions of 𝑓 to subspaces metric (∗∗) above. If 𝐸 is complete, then so is 𝐸 since 𝑋 𝑍×𝐾𝑖 are continuous, and Proposition 3 implies that the is compactly generated (see, e.g., Willard [6,Theorem 𝐾𝑖 composites 𝑅𝑖 ∘𝑓: 𝑍 →𝑌 are continuous. But as 𝑋 is 43.11]). compactly generated, a function 𝜙:𝑋is →𝑌 continuous if 𝜙| :𝐾 →𝑌 Proposition 9. Let 𝐴 be 𝑃-embedded in 𝑍 and let 𝑋 be a and only if all restrictions 𝐾𝑖 𝑖 are continuous. ∞ 𝐾 compactly generated hemicompact space. Then the subset 𝐴×𝑋 𝑍→∏ 𝑌 𝑖 This means that the obvious map 𝑖=1 ,whose is 𝑃-embedded in 𝜅(𝑍 .× 𝑋) components are 𝑅𝑖 ∘𝑓, maps into the image of the embedding 𝑋 (∗) and therefore yields a continuous function 𝑍→𝑌 AresultduetoAloandSennott(see[` 17,Theorem1.2]) which is precisely 𝑓. shows that 𝐴 is 𝑃-embedded in 𝑍 if and only if every continuous function from 𝐴 to a Frechet´ space extends Lemma 6. 𝐾 Let be a compact regular space. The topologies continuously over 𝑍. Proposition 9 seems to be the right way 𝜅((𝑍 × 𝐾) ×𝑋) 𝜅(𝑍 × 𝑋) ×𝐾 and (viewed as topologies on of generalizing the equivalence (1) ⇔ (2)ofTheorem2.4in 𝑍×𝑋×𝐾 )coincide. [17]. For a closed subset 𝐴 of 𝑍,thetopology𝜅(𝐴×𝑋) coincides We note that this is a corollary of the much more general with the topology that the set 𝐴×𝑋inherits from 𝜅(𝑍 .× 𝑋) Theorem 1.15 of Dydak [14]. (Since 𝐾 is compact regular, it For arbitrary 𝐴,thetwotopologiesmaydiffer,butnotethat is locally compact according to the definition in14 [ ].) For 𝜅(𝐴 × 𝑋) is always finer than the subspace topology. the sake of completeness, we provide an independent proof (along slightly different lines). Proof. Let 𝐸 be a Frechet´ space and let 𝑓:𝐴×𝑋be →𝐸 acontinuousmapwhere𝐴×𝑋is understood to inherit its Proof. We show that the two topologies have the same 𝜅(𝑍 × 𝑋) continuous maps into an arbitrary space 𝑌. By definition, topology from . Precomposing with the continuous 𝑓:𝜅(𝑍×𝑋×𝐾)→ 𝑌 identity 𝜅(𝐴 × 𝑋) →𝐴×𝑋 and using Lemma 5,weobtaina is continuous if and only if the 𝑋 𝑋 𝑓 : 𝑍×𝐶×𝐾 →𝑌 𝐶⊂𝑋 continuous map 𝐴→𝐸 .ByLemma 8, 𝐸 is also a Frechet´ restrictions 𝐶 (for compact ) ̂ are continuous which, by Proposition 3,isifandonlyiftheir space and as 𝐴 is 𝑃-embedded in 𝑍,thefunction𝑓 extends ̂ 𝐾 𝐹:𝑍̂ 𝑋 →𝐸 𝐹̂ adjoints 𝑓𝐶 :𝑍×𝐶→ 𝑌 are continuous. The latter is if continuously to . Reapplying Lemma 5, ̂ 𝐾 𝐹:𝜅(𝑍×𝑋)→ 𝐸 and only if the map 𝑓:𝜅(𝑍×𝑋)→𝑌 is continuous induces the desired extension . 𝑓:𝜅(𝑍×𝑋)×𝐾 →𝑌 𝑋 and this in turn if and only if Proof of Theorem 1. Let 𝑍 be metrizable and let 𝑓:𝐴 →𝑌 is continuous, by another application of Proposition 3.This be a continuous map defined on the closed subset 𝐴 of 𝑍. 𝑋 finishes the proof. By assumption, 𝑌 has the homotopy type of an ANR; hence (𝑍, 𝐴) 𝑓 admits a neighbourhood extension up to homotopy. That Let be a topological pair (no separation properties 𝑋 assumed). Then 𝐴 is 𝑃-embedded in 𝑍 if continuous pseu- is, there exist a continuous map 𝑔:𝑈 →𝑌 where 𝑈 is 𝑋 dometrics on 𝐴 extend to continuous pseudometrics on 𝑍. open and contains 𝐴 and a homotopy ℎ:𝐴×[0,1]→ 𝑌 Also, 𝐴 is a zero set in 𝑍 if there exists a continuous function beginning in 𝑔|𝐴 and ending in 𝑓. 4 Journal of Function Spaces and Applications

𝑋 Let ℎ⊔𝑔 : 𝐴×[0, 1]∪𝑈×{0} →𝑌 denote the continuous Proof. Let 𝑋 be a compactly generated hemicompact space ̂ union of the two, with adjoint ℎ⊔𝑔̂ : 𝜅((𝐴 × [0, 1]∪𝑈× with its sequence of distinguished compact subsets {𝐾𝑖} and {0}) × 𝑋) → 𝑌.As𝐴×[0,1]∪𝑈×{0}is closed in 𝐴×𝑈, let 𝑞 denote the obvious surjective map from the disjoint ̂ ∐ 𝐾 =: 𝑍 𝑋 𝐴 𝑋 the map ℎ⊔𝑔̂ is continuous with respect to the topology that union 𝑖 𝑖 to .Clearly,aset in is closed if and (𝐴×[0,1]∪𝑈×{0})×𝑋inherits from 𝜅(𝑈×[0,1]×𝑋). Under only if 𝐴∩𝐾𝑖 is closed in 𝐾𝑖 for all 𝑖.Thus,𝑞 is a quotient map. the homeomorphism 𝜅(𝑈 × [0, 1] × 𝑋) ≈ 𝜅(𝑈 × 𝑋) ×[0,1] Tautologically, 𝑍 is both 𝜎-compact and locally compact, and ̂ 𝑞 of Lemma 6,themapℎ⊔𝑔̂ corresponds to a continuous map hemicompactness renders weakly proper. ̂ For the reverse implication, suppose first that 𝑍 is a 𝑘 : 𝜅(𝐴 × 𝑋) × [0, 1] ∪ 𝜅(𝑈 × 𝑋)×{0}→𝑌. 𝐴 𝑈 𝐴×𝑋 compactly generated hemicompact space with distinguished Obviously, as is a zero set in ,theproduct is {𝐿 } 𝑞:𝑍 →𝑋 𝑈×𝑋 compact subsets 𝑖 and is a weakly a zero set in with respect to the Cartesian product 𝑞 𝑋 𝐴×𝑋 𝜅(𝑈×𝑋) proper quotient map. Clearly, as is weakly proper, is topology. A fortiori, is a zero set in . {𝑞(𝐿 )} 𝐴×𝑋 𝑃 hemicompact with distinguished compact subsets 𝑖 .To Hence, by Proposition 9,theset is 0-embedded in 𝑋 𝑓:𝑋 →𝑊 𝜅(𝑈 × 𝑋) ̂𝑘 𝐾:𝜅(𝑈× see that is compactly generated, let be a . Theorem 7 yields an extension of to function that is continuous on all compact sets. Hence, if 𝐿 𝑋) × [0, 1] →𝑌. Reapplying Lemma 6 and Lemma 5, 𝐾 𝑍 𝑓∘𝑞 𝑋 is a compact subset of , is continuous on the saturation induces a continuous function 𝑘:𝑈×[0,1]→ 𝑌 .Level 𝑞−1(𝑞(𝐿)) 𝑓 𝐿 𝑍 1 𝑓 .Consequently, is continuous on .Since is of this homotopy is a continuous extension of over the compactly generated, 𝑓∘𝑞is continuous and since 𝑞 is a 𝑈 𝑌𝑋 neighbourhood . Therefore, is an ANR. quotient map, so also is 𝑓.Sincethisholdsforall𝑓, 𝑋 is Corollary 10. 𝐶 𝑌 compactly generated. If is a compact space and is an ANR, then Finally, suppose that 𝑍 is a 𝜎-compact locally compact 𝑌𝐶 is an ANR. space. Then 𝑍 is compactly generated. (See, e.g., [6], 43.9.) 𝑍 U 𝑌𝐶 Since is locally compact, it has an open cover consisting Proof. By Theorem 3 of Milnor [3], has CW homotopy of relatively compact sets. As 𝑍 is the union of countably type. many compact sets, U has a countable subcover {𝑈𝑗|𝑗 = 1, 2, 3, . . .} 𝐿 =∪𝑖 𝑈 𝑖= Corollary 10 was proved independently by Yamashita .Thesequenceofcompactsets 𝑖 𝑗=1 𝑗, [18] (with the additional requirement that 𝐶 be Hausdorff) 1, 2, 3, . . ., exhibits 𝑍 as hemicompact. This completes the but the author of this note has not seen it elsewhere for proof. nonmetrizable compacta 𝐶.Fromthepointofviewof𝑃- embeddings, however, Corollary 10 encodes a long-known Acknowledgments fact (see Przymusinski´ [19,Theorem3]):if𝐴 is 𝑃-embedded in 𝑍 and 𝑋 is a compact space, then 𝐴×𝑋is 𝑃-embedded in The author is grateful to Atsushi Yamashita for posing the 𝑍×𝑋. question that has led to the results presented here and to 𝑋 the referee for recommending that a characterization of Proof of Corollary 2. Suppose that 𝑌 is metrizable. If 𝑌 compactly generated hemicompact spaces in the sense of contains an arc (which is if and only if it has a nontrivial 𝑋 Proposition A.1 be included in the paper. The author was path component), it follows that [0, 1] is metrizable. Since supported in part by the Slovenian Research Agency Grants 𝑋 is a Tychonoff space, points in 𝑋 canbeseparatedfrom P1-0292-0101 and J1-4144-0101. compact sets in 𝑋 by means of continuous functions 𝑋→ [0, 1]. 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