---
tag: 0PNWCM559P
kind: statement
role: proposition
name: Interior is open
msc: [54A05]
uses: [0C38S0KMJK, 01ZQQDCERP]
sources:
  - url: https://en.wikipedia.org/wiki/Interior_(topology)
    locator: Properties
---

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then the [[interior|0C38S0KMJK]] $\operatorname{int} S$ is an [[open set|01ZQQDCERP]] of $X$.

## Remarks

Proof: $\operatorname{int} S$ is a union of open sets, and each union of open sets is open by the axioms of a topology.
It follows that $\operatorname{int}(\operatorname{int} S) = \operatorname{int} S$.

## Explanations

### L1 human

The interior is made by putting together open sets, and open sets put together give an open set.