---
tag: 9HBDQ2W9CF
kind: concept
name: Completely normal space
msc: [54D15]
specializes: [0NSR9V8TED]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_space
---

## Summary

A completely normal space is a topological space in which every subspace is [[normal|0NSR9V8TED]].
Equivalently, any two separated sets, where neither meets the closure of the other, lie in disjoint open sets.
Normality is not inherited by subspaces in general, so this is a real strengthening.
A completely normal $T_1$ space is called a $T_5$ space.

## Explanations

### L1 human

Normal spaces let you pull apart two closed sets that do not meet.
In a completely normal space, you can do more: you can pull apart two sets that merely keep away from each other's closures.
For example, on the real line, the open intervals $(0, 1)$ and $(1, 2)$ touch at the point $1$, yet neither contains a limit point of the other.
They lie in the disjoint open sets $(0, 1)$ and $(1, 2)$.
Every metric space is completely normal.