---
tag: 68MA0RM5P6
kind: concept
name: Moore space
msc: ["54E30"]
specializes: [03HMJASQBQ]
sources:
  - url: https://en.wikipedia.org/wiki/Moore_space_(topology)
same_as:
  wikidata: Q6908267
  wikipedia: Moore space (topology)
---

## Summary

A Moore space is a regular Hausdorff space with a development, that is, a sequence of open covers that shrink to each point.
Every [[metrizable space|091GJ3FYD7]] is a Moore space, and Moore spaces share many properties of metric spaces.
They are the setting of the [[normal Moore space problem|Z6T00E3JQH]], a central question of set-theoretic topology.

## Explanations

### L1 human

In a metric space, the balls of radius $1, 1/2, 1/3, \dots$ around a point get smaller and close in on the point.
A Moore space keeps this picture without a distance.
It has a sequence of covers by open sets, and the sets of the $n$-th cover that contain a point close in on it as $n$ grows.
The plane with the usual topology is an example, with the cover by all open discs of radius $1/n$ as the $n$-th cover.
The [[Moore plane|SAR7PBYF07]] is a Moore space that is not metrizable.