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 | * 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]]. |
Revision as of 07:20, 29 December 2008
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
- 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.
Properties
- Compactness is a topological invariant: that is, a topological space 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 image of a compact space under a [continuous function]] into a Hausdorff space is compact.
- 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.