---
tag: MXHTKVKB09
kind: concept
name: Dowker space
msc: [54D15, 54D20, 54G20]
specializes: [0NSR9V8TED]
sources:
  - url: https://en.wikipedia.org/wiki/Dowker_space
same_as:
  wikidata: Q5302693
  wikipedia: Dowker space
---

## Summary

A Dowker space is a [[normal|0NSR9V8TED]] Hausdorff space that is not countably paracompact.
Equivalently, it is a normal Hausdorff space $X$ whose product $X \times [0, 1]$ with the unit interval is not normal.
Such spaces show that normality is not preserved by products, even with a compact metric factor.
Mary Ellen Rudin constructed a Dowker space in ZFC.

## Explanations

### L1 human

Normality is a separation property: two disjoint closed sets can be put inside disjoint open sets.
One might expect that multiplying a normal space by the plain interval $[0, 1]$ keeps it normal.
A Dowker space is a normal space for which this fails.
Every metrizable space and every [[paracompact|8R29G6NG9Q]] Hausdorff space is normal and countably paracompact, so none of them is a Dowker space.
Known Dowker spaces are hard to build, which is why they count as exotic.