---
tag: 0AJ646N339
kind: definition
name: Normal space
msc: [54D15]
defines: 0NSR9V8TED
uses: [0CNQXVBHCX, 01ZQQDCERP, 0G5AV9Q4RR]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_space
    locator: Definitions
---

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is normal when the property that follows holds.
For all disjoint [[closed sets|0G5AV9Q4RR]] $A, B \subseteq X$, there are [[open sets|01ZQQDCERP]] $U$ and $V$ with $A \subseteq U$, $B \subseteq V$ and $U \cap V = \varnothing$.

## Remarks

Names vary. Some authors call a space T4 when it is normal and [[T1|05GQJXBP3F]]. Such a space is [[regular|0JF9KW2TXB]] and [[Hausdorff|0MXB4CKHFW]].
Each [[metrizable space is normal|0AD0BZ49DW]], and each [[compact Hausdorff space is normal|0MQQQ5NA2S]].
A product of two normal spaces need not be normal. The Sorgenfrey plane is an example.
[[Urysohn's lemma|01EY3VR7ZD]] characterises normal spaces by continuous functions.

## Explanations

### L1 human

In a normal space, two closed sets that do not meet can be put into two open sets that do not meet.
This is a strong condition, but it holds for all metric spaces.