The Dowker’s theorem states that for a normal space , is normal if and only if 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 is perfectly normal, then is countably paracompact.
To prove this theorem, we use the following characterization of countably paracompactness (you can find a proof here).
Lemma. Let be a normal space. Then is countably paracompact if and only if for each sequence of closed subsets of such that and , there exist open sets such that .
Proof of Theorem. Suppose is perfectly normal. Let be a sequence of closed sets such that . For each , let where each is open in . For each , define . Clearly, . It is easy to see that . Note that all the open sets are used in defining the sequence . Thus would imply .
Comment. As a consequence of this theorem and the Dowker’s theorem, if is perfectly normal, then is normal for any compact metric space .
Dowker, C. H. , On Countably Paracompact Spaces, Canad. J. Math. 3, 219-224.
Katetov, M., On real-valued functions in topological spaces, Fund. Math. 38 (1951), pp. 85-91.
Rudin, M. E., , Dowker Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 761-780.
Revised November 23, 2016