On Spaces That Can Never Be Dowker

A Dowker space is a normal space X for which the product with the closed unit interval [0,1] is not normal. In 1951, Dowker characterized Dowker’s spaces as those spaces that are normal but not countably paracompact ([1]). Soon after, spaces that are normal but not countably paracompact became known as Dowker spaces. In 1971, M. E. Rudin ([2]) constructed a ZFC example of a Dowker’s space. But this Dowker’s space is large. It has cardinality (\omega_\omega)^\omega and is pathological in many ways. Thus the search for “nice” Dowker’s spaces continued. The Dowker’s spaces being sought were those with additional properties such as having various cardinal functions (e.g. density, character and weight) countable. Many “nice” Dowker’s spaces had been constructed using various additional set-theoretic assumptions. In 1996, Balogh constructed a first “small” Dowker’s space (cardinaltiy continuum) without additional set-theoretic axioms beyond ZFC ([4]). Rudin’s survey article is an excellent reference for Dowker’s spaces ([3]).

In this note, I make several additional observations on Dowker’s spaces. In this previous post, I presented a proof of the Dowker’s theorem characterizing the normal spaces for which the product with the unit interval is normal (see the statement of the Dowker’s theorem below). In another post, I showed that perfectly normal spaces can never be Dowker’s spaces. Based on the Dowker’s theorem, several other classes of spaces are easily seen as not Dowker.

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

  1. The space X is countably paracompact.
  2. The product X \times Y is normal for any infinite compact metric space Y.
  3. The product X \times [0,1] is normal.
  4. For each sequence of closed subsets \lbrace{A_0,A_1,A_2,...}\rbrace of X such that A_0 \supset A_1 \supset A_2 \supset ... and \bigcap_{n<\omega} A_n=\phi, there is open sets U_n \supset A_n for each n such that \bigcap_{n<\omega} U_n=\phi.

Observations. If X is perfectly compact, then it can be shown that it is countably paracompact by showing that it satisfies condition 4 in the Dowker’s theorem (there is a proof in this blog). Thus there are no perfectly normal Dowker’s spaces. There are no countably compact Dowker’s spaces since any countably compact space is countably paracompact. This can also be seen using condition 4 above. In a countably compact space, any decreasing nested sequence of closed sets has non-empty intersection and thus condition 4 is satisfied vacuously. Furthermore, all metric spaces, compact spaces, regular Lindelof spaces cannot be Dowker since these spaces are paracomapct.

Normal Moore spaces are perfectly normal. Thus there are no Dowker’s spaces that are Moore spaces. Note that a space is perfectly normal if it is normal and if every closed set is G_\delta. We show that in a Moore space, every closed set is G_\delta. Let \lbrace{\mathcal{O}_n:n \in \omega}\rbrace be a development for the regular space X. Let A be a closed set in X. We show that A is a G_\delta- set in X. For each n, let U_n=\lbrace{O \in \mathcal{O}_n:O \bigcap A \neq \phi}\rbrace. Obviously, A \subset \bigcap_n U_n. Let x \in \bigcap_n U_n. If x \notin A, there is some n such that for each O \in \mathcal{O}_n with x \in O, we have O \subset X-A. Since x \in \bigcap_n U_n, x \in O for some O \in \mathcal{O}_n and O \cap A \neq \phi, a contradiction. Thus we have A=\bigcap_n U_n.

There are other classes of spaces that can never be Dowker. We point these out without proof. For example, there are no linearly ordered Dowker’s spaces and there are no monotonically normal Dowker’s spaces (see Rudin’s survey article [3]).

Reference

  1. Dowker, C. H., On Countably Paracompact Spaces, Canad. J. Math. 3, (1951) 219-224.
  2. Rudin, M. E., A normal space X for which X \times I is not normal, Fund. Math., 73 (1971), 179-186.
  3. Rudin, M. E., Dowker Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, (1984) 761-780.
  4. Balogh, Z., A small Dowker space in ZFC, Proc. Amer. Math. Soc., 124 (1996), 2555-2560.
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