Long colimits of topological groups III: Homeomorphisms of products and coproducts

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support, or as a subgroup of the homeomorphism group of its Stone-\v{C}ech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors provide necessary and sufficient conditions for finite products of ordinals equipped with the order topology to have CSHP. In addition, necessary conditions are presented for finite products and coproducts of spaces to have CSHP.


Introduction
Given a compact space K, it is well known that the homeomorphism group Homeo(K) is a topological group with the compact-open topology ( [1]). If X is assumed to be only Tychonoff, then for every compact subset K ⊆ X, the group Homeo K (X) of homeomorphisms supported in K (i.e., identity on X\K) is a topological group with the compact-open topology; however, the full homeomorphism group Homeo(X) equipped with the compact-open topology need not be a topological group ( [5]). Nevertheless, Homeo(X) can be turned into a topological group by embedding it into Homeo(βX), the homeomorphism group of the Stone-Čech compactification of X. The latter topology has also been studied under the name of zero-cozero topology ([10], [4]).
For a Tychonoff space X, let K (X) denote the family of compact subsets of X. In light of the foregoing, the group Homeo cpt (X) := K∈K (X) Homeo K (X) of the compactly supported homeomorphisms of X admits three seemingly different topologies, listed from the finest to the coarsest: (a) the finest topology making all inclusions Homeo K (X) / / Homeo cpt (X) continuous (i.e., the colimit in the category of topological spaces and continuous functions); (b) the finest group topology making all inclusions Homeo K (X) / / Homeo cpt (X) continuous (i.e., the colimit in the category of topological groups and continuous homomorphisms); and (c) the topology induced by Homeo(βX).
Recall that the groups {Homeo K (X)} K∈K (X) are said to have the Algebraic Colimit Property (ACP) if the first and the second topologies coincide ( [2], [3]). Recall further that a space X is said to have the Compactly Supported Homeomorphism Property (CSHP) if the first and the last topologies coincide ( [2]), in which case all three topologies are equal.
In a previous work, the authors gave sufficient conditions for a finite product of ordinals to have CSHP ([2, Theorem D(c)]). The main result of this paper is that the same conditions are also necessary, thereby providing a complete characterization of CSHP among such spaces.
The proof of Theorem A is based on results of general applicability about CSHP of products and coproducts of spaces. For an infinite cardinal τ , a subset S of a space X is said to be τ -discrete in X if every subset of S of cardinality less than τ is closed in X. If S is τ -discrete in X, then every subset of S of cardinality less than τ is discrete. Being τ -discrete in X is equivalent to being closed and discrete in a certain finer topology (Proposition 2.1). Recall that the cofinality cf(I, ≤) of a partially ordered set (I, ≤) is the smallest cardinal of a cofinal set contained in I.
Theorem B. Let Y be a compact Hausdorff space, Z be a zero-dimensional locally compact Hausdorff pseudocompact space that is not compact, and τ := cf(K (Z), ⊆). If Homeo(Y ) contains a τ -discrete subset of cardinality τ that is not closed, then the product Y ×Z does not have CSHP.
Recall that the support of a homeomorphism h of a space X is supp h := cl X {x ∈ X | h(x) = x}.
Theorem C. Let Y be a compact Hausdorff space, Z a locally compact Hausdorff space, and {K α } α<τ a cofinal family in K (Z), where τ is an infinite cardinal. Suppose further that (I) Homeo(Y ) contains a τ -discrete subset of cardinality τ that is not closed; and (II) Homeo cpt (Z) contains a net (g β ) β<τ of distinct elements such that lim g β = id Z and supp g β K α whenever α < β. Then the coproduct (disjoint union) Y ∐Z does not have CSHP.
In order to invoke Theorems B and C, one needs to ensure that Homeo(Y ) contains a τ -discrete subset of cardinality τ that is not closed. For spaces that are of interest to us in this paper, this is guaranteed by the next theorem.
Theorem D. Let α be an infinite limit ordinal with τ := cf(α), and put Y = α+1 with the order topology. Then Homeo(Y ) contains a τ -discrete subset of cardinality τ that is not closed.
The paper is structured as follows. In §2, we provide some preliminary results that are used throughout the paper. In §3, we prove Theorems B and C, while the proof of Theorem D is presented in §4. Lastly, Theorem A is proven in §5.

Preliminaries
Let τ be an infinite cardinal. For a topological space (X, T ), the subsets of X of cardinality less than τ is a directed system with respect to inclusion. We put (X, T <τ ) := colim{Y | Y ⊆ X, |Y | < τ }, where the colimit is formed in the category Top of topological spaces and their continuous maps.
Proposition 2.1. Let τ be an infinite cardinal, and (X, T ) a topological space. A subset S ⊆ X is τ -discrete in X if and only if S is closed and discrete in (X, T <τ ).
PROOF. Suppose that S ⊆ X is τ -discrete. Then |S ∩Y | < τ for every Y ⊆ X with |Y | < τ , and thus S ∩Y is closed in X; in particular, S ∩Y is closed Y . Therefore, S is closed in (X, T <τ ). Let s 0 ∈ S. Then S\{s 0 } is also τ -discrete, and consequently, by the previous argument, closed in (X, T <τ ). Hence, the singleton {s 0 } is open in S in the topology induced by (X, T <τ ). This shows that S is discrete in (X, T <τ ).
Conversely, suppose that S ⊆ X is closed and discrete in (X, T <τ ). Let A ⊆ S be such that |A| < τ . We show that A is closed in (X, T ). Let y 0 ∈ X\A, and put Y := A∪{y 0 }. Then |Y | < τ , and so S ∩Y is closed and discrete in Y . If y 0 ∈ S, then S ∩Y = Y is discrete, and so A = Y \{y 0 } is closed in Y . If y 0 / ∈ S, then S ∩Y = A is closed in Y . In both cases, y 0 / ∈ cl Y A, and therefore y 0 / ∈ cl X A. This shows that A is closed in X, as desired.
The next lemma allows one to show that a space does not have CSHP by constructing a suitable τ -discrete set in its homeomorphism group. Lemma 2.3. Let X be a topological space, and {X α } α∈I a directed system of subsets of X such that X = α∈I X α . Suppose that there is an infinite cardinal τ and a subset S ⊆ X such that: (1) S is τ -discrete in X; (2) |S ∩X α | < τ for every α ∈ I; and (3) S is not closed in X.
PROOF. Let S ⊆ X be a subset with properties (1)-(3). By (1) and (2), S ∩X α is closed in X for every α ∈ I; in particular, S ∩X α is closed in X α for every α ∈ I. Thus, S is closed in colim α∈I X α .
By (3), S is not closed in X. Therefore, the two topologies are distinct.
Lastly, recall that CSHP is inherited by clopen subsets. (a) If A ⊆ X is a clopen subset and X has CSHP, then so does A. (b) If X contains an infinite discrete clopen subset, then X does not have CSHP.

Products and coproducts with compact spaces
In this section, we prove Theorems B and C. Before we prove Theorem B, we need a technical proposition about the existence of cofinal subsets with small down-sets. PROOF. Let C ⊆ I be a cofinal subset. Without loss of generality, we may assume that |C| = τ . Let C = {c α | α < τ } be an enumeration of C. We define {α γ } γ<τ inductively as follows. We put α 0 := 0. For 0 < γ < τ , suppose that α δ has already been defined for all δ < γ. We observe that {c α β } β<γ is not cofinal in I, because its cardinality is smaller than τ . Thus, Put J := {c αγ | γ < τ }. It follows from the construction of {α γ } γ<τ that c αγ c α β for every β < γ < τ.
In other words, if c αγ ≤ c α β , then γ ≤ β. Therefore, |{b ∈ J | b ≤ a}| < τ for every a ∈ J. It remains to show that J is cofinal in I. To that end, let x ∈ I. Since C is cofinal in I, the set {β < τ | x ≤ c β } is non-empty. Put δ := min{β < τ | x ≤ c β }. It follows from the construction of δ that c δ c ε for every ε < δ. ( It follows from the construction of the {α γ } γ<τ that they are strictly increasing, and in particu- For every β < γ, one has α β < δ, and thus, by Theorem B. Let Y be a compact Hausdorff space, Z be a zero-dimensional locally compact Hausdorff pseudocompact space that is not compact, and τ := cf(K (Z), ⊆). If Homeo(Y ) contains a τ -discrete subset of cardinality τ that is not closed, then the product Y ×Z does not have CSHP.
PROOF. Since Y is compact and Z is pseudocompact, the product Y ×Z is also pseudocompact ([6, 3.10.27]), and by Glicksberg's Theorem ([8, Let {C α } α<τ be a cofinal family in (K (Z), ⊆). Without loss of generality, we may assume that each C α is open in Z, and α<τ C α = ∅. Using Proposition 3.1, one may pick a cofinal subfamily We construct a subset S ⊆ G that satisfies the conditions of Lemma 2.3: This will show that G = colim α<τ G α , and thus Y ×Z does not have CSHP.
(1) Let π Y : Y ×Z → Y and π Z : Y ×Z → Z denote the respective projections, and put Since . Therefore, by (4), (3) Since f α = id Y for every α < τ , it follows that h α = id Y ×Z , and thus id Y ×Z / ∈ S. It remains to show that id Y ×Z ∈S. To that end, let W be an entourage of the diagonal in (Y ×βZ) 2 . Then for some entourage U of the diagonal in Y ×Y and entourage V of the diagonal in βZ ×βZ ([6, 8.2.1]). Since id Y ∈ S ′ , there is γ < τ such that (y, f γ (y)) ∈ U for every y ∈ Y . Therefore, (y, z, βh γ (y, z)) ∈ W for every (y, z) ∈ Y ×βZ. Hence, id Y ×Z ∈S.
Theorem C. Let Y be a compact Hausdorff space, Z a locally compact Hausdorff space, and {K α } α<τ a cofinal family in K (Z), where τ is an infinite cardinal. Suppose further that (I) Homeo(Y ) contains a τ -discrete subset of cardinality τ that is not closed; and (II) Homeo cpt (Z) contains a net (g β ) β<τ of distinct elements such that lim g β = id Z and supp g β K α whenever α < β. Then the coproduct (disjoint union) Y ∐Z does not have CSHP.
Let S ′ ⊆ Homeo(Y ) be a τ -discrete subset such that |S ′ | = τ and S ′ is not closed. Without loss of generality, we may assume that id Y ∈ S ′ \S ′ . Let S ′ = {f α | α < τ } be an injective enumeration of S ′ . For α < τ , put h α := f α ∐g α . Clearly, h α ∈ G, because Y and Z are clopen subsets of Y ∐Z.
is continuous, where the function spaces are equipped with the compact-open topology. Thus, its restriction to H and co-restriction to Homeo(Y ) ⊆ C (Y, Y ∐βZ), is a continuous group homomorphism. The restriction Γ |S is injective (because Γ(h α ) = f α ), and Γ(S) = S ′ is τ -discrete in Homeo(Y ). Therefore, by Proposition 2.2, S is τ -discrete in H.
(2) For β < τ , h β ∈ S ∩G α if and only if (supp f β )∪(supp g β ) = supp h β ⊆ Y ∪K α , or equivalently, supp g β ⊆ K α . By the assumptions on (g β ) β<τ , the latter is possible only if β ≤ α. Therefore, (3) Since f α = id Y for every α < τ , it follows that h α = id Y ∐Z , and thus id Y ∐Z / ∈ S. It remains to show that id Y ∐Z ∈S. To that end, let W be an entourage of the diagonal in (Y ∐βZ) 2 . Then there is an entourage U of the diagonal in Y ×Y and an entourage V of the diagonal in βZ ×βZ such that U ∪V ⊆ W . Since lim g β = id Z , there is α 0 < τ such that for (βg α (z), z) ∈ V for every z ∈ βZ and α ≥ α 0 . One has and thus {f α | α < α 0 } is closed, because S ′ is τ -discrete. Therefore, In particular, there is α 1 ≥ α 0 such that (f α 1 (y), y) ∈ U for every y ∈ Y . Hence, for every x ∈ Y ∐βZ, as desired.

Construction of τ -discrete subsets
Theorem D. Let α be an infinite limit ordinal with τ := cf(α), and put Y := α+1 with the order topology. Then Homeo(Y ) contains a τ -discrete subset of cardinality τ that is not closed.
The proof of Theorem D is broken down into several lemmas. First, the special case where the ordinal has countable cofinality is proven. Then, the theorem is reduced to the case where α = ω β for an infinite limit ordinal β. (Here, and throughout this paper, ω β means ordinal exponentiation, not cardinal exponentiation.) Proposition 4.1. Let α be an infinite limit ordinal, and put Y := α+1 with the order topology.
PROOF. Let U be an entourage of the diagonal ∆ Y in Y ×Y . Then U is a neighborhood of the point (α, α) ∈ U, and so there is ξ < α such that (ξ, α]×(ξ, α] ⊆ U. Let j 0 ∈ J be such that f j|[0,ξ] = id [0,ξ] for every j ≥ j 0 . Then, for every j ≥ j 0 and x ∈ Y , as desired.

Lemma 4.2. Let α be an infinite limit ordinal with countable cofinality, and put Y := α+1 with the order topology. Then Homeo(Y ) contains a countable subset that is not closed.
PROOF. Let {α n } n<ω be a strictly increasing cofinal sequence in α. Let f n : Y → Y denote the transposition Since f n is the identity for all but two isolated points, it is a homeomorphism of Y . Furthermore, lim f n = id Y in Homeo(Y ) by Proposition 4.1, because the {α n } n<ω are cofinal and increasing. Therefore, S := {f n | n < ω} is a countable subset that is not closed.
(I) By Theorem D, Homeo(Y ) contains a τ -discrete subset of cardinality τ that is not closed.

Products of ordinals
In this section, we prove Theorem A, which provides necessary and sufficient conditions for a product of ordinals to have CSHP.
Sufficiency was proven in the authors' previous work ([2, Theorem D(c)]), and so it is only necessity that has to be shown. We first prove a special case of Theorem A.
Theorem 5.1. Let λ be an infinite limit ordinal, and let X be λ equipped with the order topology. If the space X has CSHP, then λ is an uncountable regular cardinal.
PROOF. If λ had countable cofinality, then it would contain an infinite discrete clopen subset. It would follow then by Lemma 2.4(b), that λ does not have CSHP, contrary to our assumption. Thus, cf(λ) > ω.
We proceed now to prove Theorem A.
PROOF OF THEOREM A. Suppose that X has CSHP. Without loss of generality, we may assume that k ≥ 1. Put κ := min λ i . Since κ embeds as a clopen subset of X, by Lemma 2.4(a), κ has CSHP. Thus, by Theorem 5.1, κ is an uncountable regular cardinal. Put Y := κ+1 and Z := κ with the order topology. By Theorem D, Homeo(Y ) contains a κdiscrete subset of cardinality κ that is not closed. The space Z is zero-dimensional, locally compact, pseudocompact, and κ := cf(K (Z), ⊆). Therefore, by Theorem B, the product Y ×Z does not have CSHP.