---
tag: 10P0C0Y5V0
kind: definition
name: Completely normal space
msc: [54D15]
defines: 9HBDQ2W9CF
uses: [0CQHFPGJ45, 0NSR9V8TED, 0WS9P12FQR, 0QEAJB91GZ, 02BH3MCBQB, 0NW6CH38YR]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_space
---

## Definition

A [[topological space|0CQHFPGJ45]] $X$ is *completely normal* when every subset of $X$, with the [[subspace topology|0WS9P12FQR]], is a [[normal space|0NSR9V8TED]].
A completely normal space that is also [[T1|0NW6CH38YR]] is called a $T_5$ space.

## Remarks

Equivalently, $X$ is completely normal if and only if the following holds.
For all $A, B \subseteq X$ with $A \cap \overline{B} = \varnothing$ and $\overline{A} \cap B = \varnothing$, there are disjoint [[open sets|02BH3MCBQB]] $U \supseteq A$ and $V \supseteq B$.
It is enough to check that every open subspace is normal.

Example: every metric space $(X, d)$ is completely normal.
Each subspace is again a metric space, and every metric space is [[normal|0AD0BZ49DW]].

Non-example: the [[Tychonoff plank|D440JPY3QG]] is compact Hausdorff, hence normal, but it is not completely normal.
Its subspace obtained by removing one corner point is not normal.

Every [[perfectly normal space|VK3ZMD1BCP]] is completely normal.