Perfectly Normal Spaces Can Never Be Dowker Spaces

The Dowker’s theorem states that for a normal space $X$, $X \times [0,1]$ is normal if and only if $X$ is countably paracompact. Since this theorem was published, any normal space that is not countably paracompact became known as Dowker space. There are classes of spaces that can never be Dowker spaces (e.g. metrizable spaces, paracompact spaces, compact spaces and Lindelof spaces). In [Katetov], it was shown that there are no perfectly normal Dowker spaces. My blog has a proof of the Dowker’s theorem (see the proof here). For more background on Dowker’s spaces, see the survey article [Rudin]. Dowker’s theorem was published in [Dowker].

Theorem. If $X$ is perfectly normal, then $X$ is countably paracompact.

To prove this theorem, we use the following characterization of countably paracompactness (you can find a proof here).

Lemma. Let $X$ be a normal space. Then $X$ is countably paracompact if and only if for each sequence $\lbrace{A_n:n \in \omega}\rbrace$ of closed subsets of $X$ such that $A_0 \supset A_1 \supset ...$ and $\bigcap_n A_n=\phi$, there exist open sets $B_n \supset A_n$ such that $\bigcap_n B_n=\phi$.

Proof of Theorem. Suppose $X$ is perfectly normal. Let $A_0 \supset A_1 \supset ...$ be a sequence of closed sets such that $\bigcap_n A_n=\phi$. For each $n$, let $A_n=\bigcap_{i<\omega} U_{n,i}$ where each $U_{n,i}$ is open in $X$. For each $n$, define $B_n=\bigcap_{i, j \leq n}U_{i,j}$. Clearly, $B_n \supset A_n$. It is easy to see that $\bigcap_n B_n=\phi$. Note that all the open sets $U_{n,j}$ are used in defining the sequence $B_0,B_1,B_2,\cdots$. Thus $\bigcap_n B_n \neq \phi$ would imply $\bigcap_n A_n \neq \phi$.

Comment. As a consequence of this theorem and the Dowker’s theorem, if $X$ is perfectly normal, then $X \times Y$ is normal for any compact metric space $Y$.

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

[Katetov]
Katetov, M., On real-valued functions in topological spaces, Fund. Math. 38 (1951), pp. 85-91.

[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.

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$\copyright \ 2009-2016 \text{ by Dan Ma}$
Revised November 23, 2016