Compact space - Revision history

InfernalAtom683 at 08:11, 14 April 2026

2026-04-14T08:11:00Z

← Older revision		Revision as of 16:11, 14 April 2026
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	== Definition ==		== Definition ==
	A topological space <math>X</math> is compact if for every collection <math>\{U_i\}_{i\in I}</math> of opensets in <math>X</math> such that		A topological space <math>X</math> is compact if for every collection <math>\{U_i\}_{i\in I}</math> of opensets in <math>X</math> such that
	<math display="block">\bigcup_{i\in I}U_i,</math>		<math display="block">X=\bigcup_{i\in I}U_i,</math>
	there exists a finite subcollection <math>\{U_{i_1},U_{i_2},\dots,U_{i_n}\}</math> such that		there exists a finite subcollection <math>\{U_{i_1},U_{i_2},\dots,U_{i_n}\}</math> such that
	<math display="block">\bigcup_{k=1}^n U_{i_k}.</math>		<math display="block">X=\bigcup_{k=1}^n U_{i_k}.</math>

InfernalAtom683 at 22:56, 11 April 2026

2026-04-11T22:56:00Z

← Older revision		Revision as of 06:56, 12 April 2026
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	A topological space <math>X</math> is compact if for every collection <math>\{U_i\}_{i\in I}</math> of opensets in <math>X</math> such that		A topological space <math>X</math> is compact if for every collection <math>\{U_i\}_{i\in I}</math> of opensets in <math>X</math> such that
	<math display="block">\bigcup_{i\in I}U_i,</math>		<math display="block">\bigcup_{i\in I}U_i,</math>
	there exists a finite subcollection <math>\{U_{i_1},U_{i_2},\dots,\U_{i_n}\}</math> such that		there exists a finite subcollection <math>\{U_{i_1},U_{i_2},\dots,U_{i_n}\}</math> such that
	<math display="block">\bigcup_{k=1}^n U_{i_k}.</math>		<math display="block">\bigcup_{k=1}^n U_{i_k}.</math>

InfernalAtom683 at 22:55, 11 April 2026

2026-04-11T22:55:26Z

← Older revision		Revision as of 06:55, 12 April 2026
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	However, in a general topological space, a [[metric]] is typically not available, thus "boundedness" cannot be defined in a meaningful way. Therefore, an adopted definition is the one using open cover. In <math>\mathbb{R}^n</math>, this condition is equivalent to being closed and bounded, while still making sense in arbitrary topological spaces and preserving the essential properties of compact sets.		However, in a general topological space, a [[metric]] is typically not available, thus "boundedness" cannot be defined in a meaningful way. Therefore, an adopted definition is the one using open cover. In <math>\mathbb{R}^n</math>, this condition is equivalent to being closed and bounded, while still making sense in arbitrary topological spaces and preserving the essential properties of compact sets.

			== Definition ==
			A topological space <math>X</math> is compact if for every collection <math>\{U_i\}_{i\in I}</math> of opensets in <math>X</math> such that
			<math display="block">\bigcup_{i\in I}U_i,</math>
			there exists a finite subcollection <math>\{U_{i_1},U_{i_2},\dots,\U_{i_n}\}</math> such that
			<math display="block">\bigcup_{k=1}^n U_{i_k}.</math>

InfernalAtom683: InfernalAtom683 moved page Compactness to Compact space

2026-03-24T01:54:38Z

InfernalAtom683 moved page Compactness to Compact space

← Older revision		Revision as of 09:54, 24 March 2026
(No difference)

InfernalAtom683: Created page with "A '''compact''' topological space is one that behaves, in many respects, like a finite space, even if it is infinite. Specifically, a compact space is a topological space whose every open cover admits a finite subcover. Compactness is one of the most fundamental topological properties in analysis and topology. Intuitively, compactness can be understood as a generalization of being "closed and bounded"...."

2026-03-22T12:58:20Z

Created page with "A '''compact''' topological space is one that behaves, in many respects, like a finite space, even if it is infinite. Specifically, a compact space is a topological space whose every open cover admits a finite subcover. Compactness is one of the most fundamental topological properties in analysis and topology. Intuitively, compactness can be understood as a generalization of being "closed and bounded"...."

New page

A '''compact''' [[topological space]] is one that behaves, in many respects, like a finite space, even if it is infinite. Specifically, a compact space is a topological space whose every open cover admits a finite subcover. Compactness is one of the most fundamental [[Topological property|topological properties]] in [[analysis]] and [[topology]].

Intuitively, compactness can be understood as a generalization of being "[[Closed set|closed]] and [[Boundedness|bounded]]". In [[Euclidean space|Euclidean spaces]] <math>\mathbb{R}^n</math>, by the [[Heine–Borel theorem]], a set is compact if and only if it is closed and bounded.

However, in a general topological space, a [[metric]] is typically not available, thus "boundedness" cannot be defined in a meaningful way. Therefore, an adopted definition is the one using open cover. In <math>\mathbb{R}^n</math>, this condition is equivalent to being closed and bounded, while still making sense in arbitrary topological spaces and preserving the essential properties of compact sets.