---
tag: VK3ZMD1BCP
kind: concept
name: Perfectly normal space
msc: [54D15]
specializes: [9HBDQ2W9CF]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_space
---

## Summary

A perfectly normal space is a [[normal space|0NSR9V8TED]] in which every closed set is a countable intersection of open sets.
Equivalently, every closed set is the zero set of a continuous real function.
Every perfectly normal space is [[completely normal|9HBDQ2W9CF]].
Every metric space is perfectly normal.

## Explanations

### L1 human

In a perfectly normal space, each closed set can be described exactly by one continuous function: the set where the function is zero.
In a metric space, the distance to the closed set is such a function.
For example, the closed set $[0, 1]$ in $\R$ is the zero set of the distance $x \mapsto d(x, [0, 1])$.
It is also the intersection of the open sets $(-1/n, 1 + 1/n)$.