closed set
\PMlinkescapephrase
closed under
Let $(X,\tau )$ be a topological space^{}. Then a subset $C\subseteq X$ is closed if its complement $X\setminus C$ is open under the topology^{} $\tau $.
Examples:

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In any topological space $(X,\tau )$, the sets $X$ and $\mathrm{\varnothing}$ are always closed.

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Consider $\mathbb{R}$ with the standard topology. Then $[0,1]$ is closed since its complement $(\mathrm{\infty},0)\cup (1,\mathrm{\infty})$ is open (being the union of two open sets).

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Consider $\mathbb{R}$ with the lower limit topology. Then $[0,1)$ is closed since its complement $(\mathrm{\infty},0)\cup [1,\mathrm{\infty})$ is open.
Closed subsets can also be characterized as follows:
A subset $C\subseteq X$ is closed if and only if $C$ contains all of its cluster points^{}, that is, ${C}^{\prime}\subseteq C$.
So the set $\{1,1/2,1/3,1/4,\mathrm{\dots}\}$ is not closed under the standard topology on $\mathbb{R}$ since $0$ is a cluster point not contained in the set.
Title  closed set 

Canonical name  ClosedSet 
Date of creation  20130322 12:30:23 
Last modified on  20130322 12:30:23 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  10 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 5400 
Synonym  closed subset 
Defines  closed 