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

单词	Closure
释义	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 $\\overline{A}^{Y}=(0,1)$ .
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