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. | |||
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]]. | |||
== | ==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 | |||
:<math>\mathcal{U}=\{A_{\gamma} \mid A_{\gamma} \subset X,\,\gamma \in \Gamma \},</math> | |||
where <math>\Gamma</math> is an arbitrary index set, and satisfies | |||
:<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> | |||
==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). | |||
[[ | ==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>. | |||
[[ | ==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 [[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
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
- 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 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.
- 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.