## Tsukuba Journal of Mathematics

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

### Dowker spaces revisited

Lewis D. Ludwig, Peter Nyikos, and John E. Porter

#### 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

**Subjects**

Primary: 54D15: Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)

**Keywords**

$\alpha$-normal $\beta$-normal products Dowker Moore spaces $\alpha$-countably paracompact

#### 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