---
tag: 8M0QH4Z60S
kind: definition
name: Regular open set
msc: [54A05]
defines: 833XAA2Q65
uses: [0CQHFPGJ45, 02BH3MCBQB, 0QEAJB91GZ, 0RZH6W98HK]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_open_set
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
For $S \subseteq X$, write $\overline{S}$ for the [[closure|0QEAJB91GZ]] of $S$ and $\operatorname{int} S$ for its [[interior|0RZH6W98HK]].
A subset $U \subseteq X$ is *regular open* if
$$U = \operatorname{int} \overline{U}.$$

## Remarks

A regular open set is [[open|02BH3MCBQB]], since an interior is always open.
For every open $U$, one has $U \subseteq \operatorname{int} \overline{U}$, and the set $\operatorname{int} \overline{U}$ is regular open.

Example: in $\R$, let $U = (0, 1)$.
Then $\overline{U} = [0, 1]$ and $\operatorname{int} [0, 1] = (0, 1) = U$, so $U$ is regular open.

Non-example: in $\R$, let $U = (0, 1) \cup (1, 2)$.
Then $\overline{U} = [0, 2]$ and $\operatorname{int} \overline{U} = (0, 2) \ne U$, so $U$ is open but not regular open.

Dually, a set $F$ is *regular closed* if $F = \overline{\operatorname{int} F}$.
A set is regular open if and only if its complement is regular closed.