Dowker’s Theorem

Let \mathbb{I} be the closed unit interval [0,1]. In 1951, Dowker proved that for normal spaces X, X \times \mathbb{I} is normal if and only if X is countably paracompact. A space X is countably paracompact if every countable open cover of X has a locally finite open refinement. Dowker’s theorem is a fundamental result on products of normal spaces. With this theorem, a question was raised about the existence of a normal but not countably paracompact space (such a space became known as a Dowker space). For a detailed discussion on Dowker spaces, see the survey paper [Rudin]. The focus here is on presenting a proof for the Dowker’s theorem, laying the groundwork for future discussion.

An open cover \mathcal{V}=\lbrace{V_\alpha:\alpha \in S}\rbrace of a space X is shrinkable if there exists an open cover\mathcal{W}=\lbrace{W_\alpha:\alpha \in S}\rbrace such that \overline{W_\alpha} \subset V_\alpha for each \alpha \in S. The open cover \mathcal{W} is called a shrinking of \mathcal{V}. We use the following lemma in proving Dowker’s theorem. Go here to see a proof of this lemma.

Lemma
A space X is normal if and only if every point-finite open cover of X is shrinkable.

Dowker’s Theorem
For a normal space X, the following conditions are equivalent:

  1. X is a countably paracompact space.
  2. If \lbrace{U_n:n<\omega}\rbrace is an open cover of X, then there is a locally finite open refinement \lbrace{V_n:n<\omega}\rbrace such that \overline{V_n} \subset U_n for each n.
  3. X \times Y is normal for any compact metrizable space Y.
  4. X \times \mathbb{I} is normal.
  5. For each sequence of closed sets such that A_0 \supset A_1 \supset ... and \cap_n A_n=\phi, there exists open sets B_n \supset A_n such that \cap_n B_n=\phi.

Proof
1 \Longrightarrow 2
Suppose X is countably paracompact. Let \mathcal{U}=\lbrace{U_n:n<\omega}\rbrace be an open cover of X. Then \mathcal{U} has a locally open refinement \mathcal{V}. For each n<\omega, let W_n=\bigcup \lbrace{V \in \mathcal{V}:V \subset U_n}\rbrace. Note that \lbrace{W_n:n<\omega}\rbrace is still locally finite (thus point-finite). By the lemma, it has a shrinking \lbrace{V_n:n<\omega}\rbrace.

2 \Longrightarrow 3
Let Y be a compact metrizable space. Let \lbrace{E_0,E_1,E_2,...}\rbrace be a base of Y such that it is closed under finite unions. Let H,K be two disjoint closed subsets of X \times Y. The goal here is to find an open set \mathcal{V} such that H \subset \mathcal{V} and \overline{\mathcal{V}} misses K.

For each x \in X, consider the following:
S_x=\lbrace{y \in Y: (x,y) \in H}\rbrace
T_x=\lbrace{y \in Y: (x,y) \in K}\rbrace.

For each n \in \omega, let O_n be defined by:
O_n=\lbrace{x \in X:S_x \subset E_n} and T_x \subset Y-\overline{E_n}\rbrace.

Claim 1.
Each x \in X is an element of some O_n.

Both S_x and T_x are compact. Thus we can find some E_n such that S_x \subset E_n and T_x \subset Y-\overline{E_n}.

Claim 2.
Each O_n is open in X.

Fix z \in O_n. For each y \in Y-E_n, (z,y) \notin H. Choose open set A_y \times B_y such that (z,y) \in A_y \times B_y and A_y \times B_y misses H. The set of all B_y is a cover of Y-E_n. We can find a finite subcover covering Y-E_n, say B_{y(0)},B_{y(1)},...,B_{y(k)}. Let A=\bigcap_i A_{y(i)}.

For each a \in \overline{E_n}, (z,a) \notin K. Choose open set C_a \times D_a such that (z,a) \in C_a \times D_a and C_a \times D_a misses K. The set of all D_a is a cover of \overline{E_n}. We can find a finite subcover covering \overline{E_n}, say D_{a(0)},D_{a(1)},...,D_{a(j)}. Let C=\bigcap_i C_{a(i)}.

Now, let O=A \cap C. Clearly z \in O. To show that O \subset O_n, pick x \in O. We want to show S_x \subset E_n and T_x \subset Y-\overline{E_n}. Suppose we have y \in S_x and y \in Y-E_n. Then y \in B_{y(i)} for some i. As a result, (x,y) \in A_{y(i)} \times B_{y(i)}. This would mean that (x,y) \notin H. But this contradicts with y \in S_x. So we have S_x \subset E_n. On the other hand, suppose we have y \in T_x and y \in \overline{E_n}. Then y \in D_{a(i)} for some i. As a result, (x,y) \in C_{a(i)} \times D_{a(i)}. This means that (x,y) \notin K. This contradicts with y \in T_x. So we have T_x \subset Y-\overline{E_n}. It follows that O \subset O_n and O_n is open in X.

Claim 3.
We can find an open set \mathcal{V} such that H \subset \mathcal{V} and \overline{\mathcal{V}} misses K.

Let \mathcal{O}=\bigcup_n O_n \times \overline{E_n}. Note that H \subset \mathcal{O} and \mathcal{O} misses K. The open set \mathcal{V} being constructed will satisfiy H \subset \mathcal{V} \subset \overline{\mathcal{V}} \subset \mathcal{O}. The open cover \lbrace{O_n}\rbrace has a locally finite open refinement \lbrace{V_n}\rbrace such that \overline{V_n} \subset O_n for each n. Now define \mathcal{V}=\bigcup_n V_n \times E_n. Note that H \subset \mathcal{V}.

We also have \overline{\mathcal{V}}=\overline{\bigcup_n V_n \times E_n}=\bigcup_n \overline{V_n \times E_n}. We have a closure preserving situation because the open sets V_n are from a locally finite collection. Continue the derivation and we have:

=\bigcup_n \overline{V_n} \times \overline{E_n} \subset \bigcup_n O_n \times \overline{E_n}=\mathcal{O}.

3 \Longrightarrow 4 is obvious.

4 \Longrightarrow 5
Let A_0 \supset A_1 \supset ... be closed sets with empty intersection. We show that A_n can be expanded by open sets B_n such that A_n \subset B_n and \bigcap_n B_n=\phi.

Choose p \in \mathbb{I} and a sequence of distinct points p_n \in \mathbb{I} converging to p. Let H=\cup \lbrace{A_n \times \lbrace{p_n}\rbrace:n \in \omega}\rbrace and K=X \times \lbrace{p}\rbrace. These are disjoint closed sets. Since X \times \mathbb{I} is normal, we can find open set V \subset X \times \mathbb{I} such that H \subset V and \overline{V} misses K.

Let B_n=\lbrace{x \in X:(x,p_n) \in V}\rbrace. Note that each B_n is open in X. Also, A_n \subset B_n. We want to show that \bigcap_n B_n=\phi. Let x \in X. The point (x,p) \in K and (x,p) \notin \overline{V}. There exist open E \subset X and open F \subset Y such that (x,p) \in E \times F and (E \times F) \cap \overline{V}=\phi. Then p_n \in F for some n. Note that (x,p_n) \notin V. Thus x \notin B_n. It follows that \bigcap_n B_n=\phi.

5 \Longrightarrow 1
Let \lbrace{T_n:n \in \omega}\rbrace be an open cover of X. Let E_n=\bigcup_{i \leq n} T_i. Let A_n=X-E_n. Each A_n is closed and \bigcap_n A_n=\phi. So there exist open sets B_n \supset A_n such that \bigcap_n B_n=\phi.

Since X is normal, there exists open W_n such that A_n \subset W_n \subset \overline{W_n} \subset B_n. It follows that \bigcap_n \overline{W_n}=\phi. Let O_n=X-\bigcap_{i \leq n} \overline{W_n}.

Claim. \lbrace{O_n:n \in \omega}\rbrace is an open cover of X. Furthermore, \overline{O_n} \subset \bigcup_{i \leq n}T_i.
Let x \in X. Since \bigcap_n B_n=\phi, x \notin B_n for some n. Then x \notin \overline{W_n} and x \in O_n. To show the second half of the claim, let y \notin \bigcup_{i \leq n}T_i. Then y \notin E_i for each i \leq n. This means y \in A_i and y \in W_i for each i \leq n. Then \bigcap_{i \leq n}W_i is an open set containing y that contains no point of O_n. Thus we have \overline{O_n} \subset \bigcup_{i \leq n}T_i.

Let S_n=T_n-\bigcup_{i < n} \overline{O_i}. We show that \lbrace{S_n:n \in \omega}\rbrace is a locally finite open refinement of \lbrace{T_n:n \in \omega}\rbrace. Clearly S_n \subset T_n. To show that the open sets S_n form a cover, let x \in X. Choose least n such that x \in T_n. Then for each i<n, x \notin E_i. This means x \in A_i and x \in \overline{W_i} for each i<n. It follows that x \notin O_i for each i<n. This implies x \in S_n. To see that the open sets S_n form a locally finite collection, note that each x \in X belong to some O_n. The open set O_n misses S_m for all m>n. Thus O_n can only meet S_i for i \leq n. We just prove that X is countably paracompact.

Note. In 4 \Longrightarrow 5, all we need is that the factor Y has a non-trivial convergent sequence. That is, if X \times Y is normal and Y has a non-trivial convergent sequence, then X satisfies condition 5.

Reference
[Dowker] Dowker, C. H. [1951], On Countably Paracompact Spaces, Canad. J. Math. 3, 219-224.

[Rudin] Rudin, M. E., [1984], Dowker Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 761-780.

____________________________________________________________________
\copyright \ 2009-2016 \text{ by Dan Ma}
Revised November 27, 2016

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