closureClosure2002-01-03 21:06:222006-12-08 08:38:57Definitiontopology\usepackage{amssymb}
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\usepackage{xypic}The \emph{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 $\overline{A}^Y=(0,1)$.