---
tag: 0SHC6Z304Z
kind: statement
role: proposition
name: Closure is the smallest closed superset
msc: [54A05]
uses: [0DRKT1QJR7, 02TZDKG50Q]
sources:
  - url: https://en.wikipedia.org/wiki/Closure_(topology)
    locator: Definitions
---

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then $S \subseteq \overline{S}$, the set $\overline{S}$ is closed, and each closed set $F$ of $X$ with $S \subseteq F$ satisfies $\overline{S} \subseteq F$.

## Remarks

Here $\overline{S}$ is the [[closure|0DRKT1QJR7]] of $S$. That it is closed is the statement [[closure is closed|02TZDKG50Q]].
This is the dual of the statement that the [[interior is the largest open subset|0HAZEFH137]].
If $S \subseteq T$, then $\overline{S} \subseteq \overline{T}$.

## Explanations

### L1 human

Among all closed sets that contain $S$, the closure is the smallest one.
For example, each closed set that contains $(0, 1)$ in $\mathbb{R}$ also contains $[0, 1]$.