---
tag: E0A14ZEW72
kind: definition
name: Dowker space
msc: [54D15, 54D20, 54G20]
defines: MXHTKVKB09
uses: [0NSR9V8TED, 0S1FJ8VEX9, DZ297MMGV8, 8R29G6NG9Q]
sources:
  - url: https://en.wikipedia.org/wiki/Dowker_space
---

## Definition

Let $X$ be a topological space.
$X$ is *countably paracompact* if every countable open cover of $X$ has an open refinement that is a [[locally finite family|DZ297MMGV8]].
$X$ is a *Dowker space* if:

1. $X$ is [[Hausdorff|0S1FJ8VEX9]],
2. $X$ is [[normal|0NSR9V8TED]], and
3. $X$ is not countably paracompact.

## Remarks

By a theorem of Dowker, a normal space $X$ is countably paracompact if and only if the product $X \times [0, 1]$ is normal.
So a Dowker space is the same as a normal Hausdorff space $X$ with $X \times [0, 1]$ not normal.
Another equivalent condition for a normal space to be countably paracompact: for every decreasing sequence $F_1 \supseteq F_2 \supseteq \cdots$ of closed sets with empty intersection there are open sets $U_n \supseteq F_n$ with empty intersection.
An example is the Dowker space constructed by Rudin in ZFC; the check of its properties is long.
A non-example is the real line $\R$: it is metrizable, hence [[paracompact|8R29G6NG9Q]], hence countably paracompact, so clause 3 fails.