Compact space: Difference between revisions

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In [[mathematics]], a compact space is a [[topological space]] for which every covering of that space by a collection of [[open set]]s has a finite subcovering. If the space is a [[metric space]] then compactness is equivalent to the set being [[completeness|complete]] and [[totally bounded set|totally bounded]] and again equivalent to [[sequential compactness]]: that every sequence in the set has a convergent subsequence.  
In [[mathematics]], a '''compact space''' is a [[topological space]] for which every covering of that space by a collection of [[open set]]s has a finite subcovering. If the space is a [[metric space]] then compactness is equivalent to the set being [[completeness|complete]] and [[totally bounded set|totally bounded]] and again equivalent to [[sequential compactness]]: that every sequence in the set has a convergent subsequence.  


A subset of a topological space is compact if it is compact with respect to the [[subspace topology]].   
A subset of a topological space is compact if it is compact with respect to the [[subspace topology]].   
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where <math>\Gamma</math> is an arbitrary index set, and satisfies
where <math>\Gamma</math> is an arbitrary index set, and satisfies
:<math>A \subset \bigcup_{\gamma \in \Gamma }A_{\gamma}.</math>  
:<math>A \subset \bigcup_{\gamma \in \Gamma }A_{\gamma}.</math>  
An '''open cover''' is a cover in which all of the sets <math>A_\gamma</math> are open. Finally, a '''subcover''' of <math>\mathcal{U}</math> is a subset <math>\mathcal{U}' \subset \mathcal{U}</math> of the form  
An '''[[open cover]]''' is a cover in which all of the sets <math>A_\gamma</math> are open. Finally, a '''subcover''' of <math>\mathcal{U}</math> is a subset <math>\mathcal{U}' \subset \mathcal{U}</math> of the form  
:<math>\mathcal{U}'=\{A_{\gamma} \mid A_{\gamma} \subset X,\,\gamma \in \Gamma'\}</math>  
:<math>\mathcal{U}'=\{A_{\gamma} \mid A_{\gamma} \subset X,\,\gamma \in \Gamma'\}</math>  
with <math>\Gamma' \subset \Gamma</math> such that  
with <math>\Gamma' \subset \Gamma</math> such that  
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==Properties==
==Properties==
* Compactness is a [[topological invariant]]: that is, a topolgical space [[homeomorphism|homeomorphic]] to a compact space is again compact.
* Compactness is a [[topological invariant]]: that is, a topological space [[homeomorphism|homeomorphic]] to a compact space is again compact.
* A [[closed set]] in a compact space is again compact.
* A [[closed set]] in a compact space is again compact.
* A subset of a [[Hausdorff space]] which is compact (with the [[subspace topology]]) is closed.
* A subset of a [[Hausdorff space]] which is compact (with the [[subspace topology]]) is closed.
* The image of a compact space under a continuous function is compact.
* The image of a compact space under a [continuous function]] into a Hausdorff space is compact.
** A continuous [[real number|real]]-valued function on a compact space is [[bounded set|bounded]] and attains its bounds.
* The [[Cartesian product]] of two (and hence finitely many) compact spaces with the [[product topology]] is compact.
* The ''[[Tychonoff product theorem]]'': The product of any family of compact spaces with the product topology is compact.  This is equivalent to the [[Axiom of Choice]].

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In mathematics, a compact space is a topological space for which every covering of that space by a collection of open sets has a finite subcovering. If the space is a metric space then compactness is equivalent to the set being complete and totally bounded and again equivalent to sequential compactness: that every sequence in the set has a convergent subsequence.

A subset of a topological space is compact if it is compact with respect to the subspace topology. A compact subset of a Hausdorff space is closed, but the converse does not hold in general. For the special case that the set is a subset of a finite dimensional normed space, such as the Euclidean spaces, then compactness is equivalent to that set being closed and bounded: this is the Heine-Borel theorem.

Cover and subcover of a set

Let A be a subset of a set X. A cover for A is any collection of subsets of X whose union contains A. In other words, a cover is of the form

where is an arbitrary index set, and satisfies

An open cover is a cover in which all of the sets are open. Finally, a subcover of is a subset of the form

with such that

Formal definition of compact space

A topological space X is said to be compact if every open cover of X has a finite subcover, that is, a subcover which contains at most a finite number of subsets of X (in other words, the index set is finite).

Examples

Properties