# 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 $\overline{A}^{Y}=(0,1)$.

Title closure Closure 2013-03-22 12:05:40 2013-03-22 12:05:40 mathwizard (128) mathwizard (128) 9 mathwizard (128) Definition msc 54A99 ClosureAxioms Interior