Closure

Definition

Let $X$ be a topological space and let $S \subseteq X$. The closure of $S$ is the intersection of all closed sets $F$ of $X$ with $S \subseteq F$. It is written $\overline{S}$ or $\operatorname{cl} S$.

Remarks

The family of these closed sets is not empty, because it contains $X$. A point $x$ is in $\overline{S}$ if and only if each open set that contains $x$ meets $S$. In $\mathbb{R}$, the closure of $(0, 1)$ is $[0, 1]$, and the closure of $\mathbb{Q}$ is $\mathbb{R}$. The closure and the interior are related by $X \setminus \overline{S} = \operatorname{int}(X \setminus S)$.

Explanations

L1 human

The closure of a set adds to the set all the points that you can approach from inside it. For the open interval $(0, 1)$, you add the two end points $0$ and $1$.