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# closure

The *closure* $\overline{A}$ of a subset $A$ of a topological space $X$ is the intersection of all closed sets containing $A$.

Equivalently, $\overline{A}$ consists of $A$ together with all limit points of $A$ in $X$ or equivalently $x\in\overline{A}$ if and only if every neighborhood of $x$ intersects $A$. Sometimes the notation $\operatorname{cl}(A)$ is used.

If it is not clear, which topological space is used, one writes $\overline{A}^{X}$. Note that if $Y$ is a subspace of $X$, then $\overline{A}^{X}$ may not be the same as $\overline{A}^{Y}$. For example, if $X=\mathbb{R}$, $Y=(0,1)$ and $A=(0,1)$, then $\overline{A}^{X}=[0,1]$ while $\overline{A}^{Y}=(0,1)$.

Keywords:

topology

Related:

ClosureAxioms, Interior

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

54A99*no label found*

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