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In [[mathematics]], a compact set is a [[set]] for which every covering of that set by a collection of sets has a finite subcovering. If the set is a subset of a [[metric space]] then compactness is equivalent to the set being [[closed set|closed]] and [[totally bounded set|totally bounded]] or, equivalently, that every sequence in the set has a convergent subsequence. For the special case that the set is a subset of a finite dimensional [[vector space]], such as the Euclidean spaces, then compactness is equivalent to that set being closed and [[bounded set|bounded]].
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==Cover and subcover of a set==
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.  
Let ''A'' be a subset of a set ''X''. A '''cover''' for ''A'' is any collection of sets of the form <math>\mathcal{U}=\{A_{\gamma} \mid A_{\gamma} \subset X,\,\gamma \in \Gamma \}</math> , where <math>\Gamma</math> is an arbitrary index set, such that <math>A \subset \cup_{\gamma \in \Gamma }A_{\gamma}</math>. For any such cover <math>\mathcal{U}</math>, a set <math>\mathcal{U}' \subset \mathcal{U}</math> of the form <math>\mathcal{U}'=\{A_{\gamma} \mid A_{\gamma} \subset X,\,\gamma \in \Gamma'\}</math> with <math>\Gamma' \subset \Gamma</math> and such that <math>A \subset \cup_{\gamma \in \Gamma'}A_{\gamma}</math> is said to be a '''subcover''' of <math>\mathcal{U}</math>.  


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 space]]s, then compactness is equivalent to that set being closed and [[bounded set|bounded]]: this is the [[Heine-Borel theorem]].


==Formal definition of compact set==
==Cover and subcover of a set==
A subset ''A'' of a set ''X'' is said to be '''compact''' if ''every'' cover of ''A'' has a ''finite'' subcover, that is, a subcover which contains at most a finite number of subsets of ''X'' (or has a finite index 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
 
:<math>\mathcal{U}=\{A_{\gamma} \mid A_{\gamma} \subset X,\,\gamma \in \Gamma \},</math> 
==See also==
where <math>\Gamma</math> is an arbitrary index set, and satisfies
[[Open set]]
:<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
:<math>\mathcal{U}'=\{A_{\gamma} \mid A_{\gamma} \subset X,\,\gamma \in \Gamma'\}</math>
with <math>\Gamma' \subset \Gamma</math> such that
:<math>A \subset \bigcup_{\gamma \in \Gamma'}A_{\gamma}.</math>


[[Closed set]]
==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 <math>\Gamma'</math> is finite).


[[Topological space]]
==Finite intersection property==
Just as the topology on a topological space may be defined in terms of the [[closed set]]s rather than the [[open set]]s, so we may transpose the definition of compactness in terms of open sets into a definition in terms of closed sets.  A space is compact if the closed sets have the ''finite intersection property'': if <math>\{ F_\lambda : \lambda \in \Lambda \}</math> is a family of closed sets with [[empty set|empty]] intersection, <math>\bigcap_{\lambda \in \Lambda} F_\lambda = \emptyset</math>, then a finite subfamily <math>\{ F_{\lambda_i} : i=1,\ldots,n \}</math> has empty intersection <math>\bigcap_{i=1}^n F_{\lambda_i} = \emptyset</math>.


[[Category:Mathematics_Workgroup]]
==Examples==
* Any finite space.
* An [[indiscrete space]].
* A space with the [[cofinite topology]].
* The ''[[Heine-Borel theorem]]'': In [[Euclidean space]] with the usual topology, a [[subset]] is compact if and only if it is closed and bounded.


[[Category:CZ Live]]
==Properties==
* 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 subset of a [[Hausdorff space]] which is compact (with the [[subspace topology]]) is closed.
* The [[quotient topology]] on an image of a compact space is compact
* The image of a compact space under a [[continuous map]] to 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]].
* If a space is both compact and Hausdorff then no finer topology on the space is compact, and no coarser topology is Hausdorff.

Revision as of 14:44, 30 December 2008

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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).

Finite intersection property

Just as the topology on a topological space may be defined in terms of the closed sets rather than the open sets, so we may transpose the definition of compactness in terms of open sets into a definition in terms of closed sets. A space is compact if the closed sets have the finite intersection property: if is a family of closed sets with empty intersection, , then a finite subfamily has empty intersection .

Examples

Properties