Homotopy Characterization of ANR Function Spaces

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 ?̂?, there exists a retraction of an open neighbourhood onto ?̂?. 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 Y, 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 Y is an ANR ifX is a metrizable compactum. For a negative example, consider the discrete space of natural numbers N and the two-point discrete ANR {0, 1}. Then {0, 1} is a Cantor set and hence certainly not an ANR. In fact, as the path components of {0, 1} 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 Y 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, undermild restrictions onX, ifY is an ANR andY is a metrizable space with the homotopy type of a CW complex, Y is in fact an ANR. Put another way, Y is an ANR if and only if Y is an ANR and Y is a metrizable space with the homotopy type of an ANR.


Introduction
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 Ŷ, there exists a retraction of an open neighbourhood onto Ŷ.We refer the reader to the first part of Mardešić [1] 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 N 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   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 {0, 1, 1/2, 1/3, . ..}.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 [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 , 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 [5] 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 [6]).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 [7] 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  0 -embeddings (see Morita [8]).
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.

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 P 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 [12]).Consequently, if   denotes the supmetric on    induced by a metric on , then   is metrizable by the metric Proposition 3. Let , , and  be topological spaces.Let  :  →   be any function with set-theoretic adjoint f :  ×  → .If f 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 f 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 f if  is locally compact and  is regular, as it is apparently not so standard.
Suppose that  is continuous and f( 0 ,  0 ) =:  0 lies in the open set  ⊂ .As  is regular, there is an open set  with  0 ∈  ⊂  ⊂ .As  is locally compact,  0 is contained in the interior of a compact set . Write  = ( 0 ).Clearly,  =  −1 ()∩ is a compact set contained in  −1 ().This means that ( 0 ) lies in the open set (, ).As  is continuous, there is an open neighbourhood  of  0 so that () ⊂ (, ).Consequently, f( × ) ⊂ .As  0 lies in the interior of , f is continuous at ( 0 ,  0 ).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 [14], ( × ) 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 f :  ×  → .Then  is continuous if and only if f : ( × ) →  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 f| ×  :  ×   → .As each compact set  is contained in one of the   , it follows by definition of ( × ) that f : ( × ) →  is continuous.
For the converse, assume that f : ( × ) →  is continuous.This means that the restrictions of f 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  → ∏ ∞ =1    , 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 [14].(Since  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 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 f :  ×  →   are continuous.The latter is if and only if the map f : ( × ) →   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 For example, every closed subset of a metrizable space is  0 -embedded.
We need -embeddings in the context of Morita's homotopy extension theorem (which in fact characterizes ANR spaces; see Stramaccia [15]).
Theorem 7 (Morita [8]).If  is  0 -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  : ×{0} →  and ℎ : ×[0, 1] →  are continuous maps that agree pointwise on  × {0}, then there exists a continuous map  :  × [0, 1] →  extending both  and ℎ.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 0 in , constitute a convex local base for   (see Schaefer [16], 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 [17].
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 f extends continuously to F :  →   .Reapplying Lemma 5, F 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 ℎ :  × [0, 1] →   beginning in |  and ending in .
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  0 -embedded in ( × ).Theorem 7 yields an extension of k to  : ( × ) × [0, 1] → .Reapplying Lemma 6 and Lemma 5,  induces a continuous function  :  × [0, 1] →   .Level 1 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 [3],   has CW homotopy type.
Corollary 10 was proved independently by Yamashita [18] (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 [0, 1]  is metrizable.Since  is a Tychonoff space, points in  can be separated from compact sets in  by means of continuous functions  → [0, 1].The proof of Theorem 8 of Arens [5] can be adapted almost verbatim to render  hemicompact.The statement of Corollary 2 follows immediately from Theorem 1.
(|   , |   )} .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.
[5, * ) (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 .