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
Canonical name	Closure
Date of creation	2013-03-22 12:05:40
Last modified on	2013-03-22 12:05:40
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	9
Author	mathwizard (128)
Entry type	Definition
Classification	msc 54A99
Related topic	ClosureAxioms
Related topic	Interior