## Tsukuba Journal of Mathematics

### Dowker spaces revisited

#### Abstract

In 1951, Dowker proved that a space $X$ is countably paracompact and normal if and only if $X \times {\bf I}$ is normal. A normal space $X$ is called a Dowker space if $X \times {\bf I}$ is not normal. The main thrust of this article is to extend this work with regards $\alpha$-normality and $\beta$-normality. Characterizations are given for when the product of a space $X$ and $(\omega + 1)$ is $\alpha$-normal or $\beta$-normal. A new definition, $\alpha$-countably paracompact, illustrates what can be said if the product of $X$ with a compact metric space is $\beta$-normal. Several examples demonstrate that the product of a Dowker space and a compact metric space may or may not be $\alpha$-normal or $\beta$-normal. A collectionwise Hausdorff. Moore space constructed by M. Wage is shown to be $\alpha$-normal but not $\beta$-nornal.

#### Article information

Source
Tsukuba J. Math., Volume 34, Number 1 (2010), 1-11.

Dates
First available in Project Euclid: 8 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.tkbjm/1283967404

Digital Object Identifier
doi:10.21099/tkbjm/1283967404

Mathematical Reviews number (MathSciNet)
MR2723720

Zentralblatt MATH identifier
1203.54015

#### Citation

Ludwig, Lewis D.; Nyikos, Peter; Porter, John E. Dowker spaces revisited. Tsukuba J. Math. 34 (2010), no. 1, 1--11. doi:10.21099/tkbjm/1283967404. https://projecteuclid.org/euclid.tkbjm/1283967404