K-Dowker spaces

Summary

INTRODUCTION

By a space we mean a Hausdorff topological space.

In one of the most referred to papers in all of mathematics, Dowker (1951), we have:

Theorem 1

The following are equivalent for a normal space X:

1. (1) X is countably paracompact.

2. (2) X × (ω + 1) is normal.

3. (3) Every countable open cover of X can be shrunk.

Part of Dowker's theorem was that X × (ω + 1) is normal if and only if X × I is normal for the closed unit interval I, in fact, if and only if X × C is normal for all compact metric C. At that time a space was said to be binormal if not only X but also X × I was normal, a hypothesis used in a number of homotopy extension theorems (see Borsuk (1937), Morita (1975), Starbird (1975)). The normality of products was a subject full of mysteries and a well known problem was to try to find a normal nonbinormal space. After Dowker's theorem such a space, if any, became known as a Dowker space.

Paracompactness was then a rather new concept, and the idea of using cardinal functions in topology, for instance studying nonparacompact spaces in terms of the minimal cardinality of an open cover having no locally finite refinement, was not so common as it is today.

As Dowker recognized, condition (3), although not the countable case of some clearly important concept, is the useful condition.