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<p><b>New page</b></p><div>[[File:Compact.svg|Compact|thumb]] '''Compact space'''<br />
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In [[mathematics]], particularly in [[topology]], a '''compact space''' is a type of [[topological space]] that, in a certain sense, is limited in size. The concept of compactness is a generalization of the notion of a closed and bounded subset of [[Euclidean space]].<br />
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== Definition ==<br />
A topological space \( X \) is said to be '''compact''' if every open cover of \( X \) has a finite subcover. This means that for any collection of open sets whose union includes \( X \), there exists a finite number of these open sets that still cover \( X \).<br />
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== Examples ==<br />
* The closed interval \([0, 1]\) in the [[real number|real numbers]] with the standard topology is a compact space.<br />
* Any finite space is compact.<br />
* The [[Cantor set]] is an example of a compact space.<br />
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== Properties ==<br />
* A compact space is always [[Lindelöf space|Lindelöf]] and [[paracompact space|paracompact]].<br />
* In a [[Hausdorff space]], compact subsets are closed.<br />
* The [[continuous image]] of a compact space is compact.<br />
* A compact subset of a [[metric space]] is [[sequentially compact]].<br />
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== Compactness in Metric Spaces ==<br />
In the context of [[metric spaces]], a subset is compact if and only if it is both [[sequentially compact]] and [[totally bounded]].<br />
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== Related Concepts ==<br />
* [[Sequential compactness]]<br />
* [[Limit point compactness]]<br />
* [[Local compactness]]<br />
* [[Paracompact space]]<br />
* [[Lindelöf space]]<br />
* [[Tychonoff theorem]]<br />
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== Applications ==<br />
Compact spaces are fundamental in various areas of mathematics, including [[analysis]], [[algebraic topology]], and [[functional analysis]]. They are used in the study of [[continuous functions]], [[differential equations]], and [[measure theory]].<br />
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== See Also ==<br />
* [[Topological space]]<br />
* [[Metric space]]<br />
* [[Hausdorff space]]<br />
* [[Open cover]]<br />
* [[Closed set]]<br />
* [[Bounded set]]<br />
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== Related Pages ==<br />
* [[Topological space]]<br />
* [[Metric space]]<br />
* [[Hausdorff space]]<br />
* [[Open cover]]<br />
* [[Closed set]]<br />
* [[Bounded set]]<br />
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[[Category:Topology]]<br />
[[Category:Mathematical analysis]]<br />
[[Category:Mathematical concepts]]<br />
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