Compact space: Difference between revisions
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* A space with the [[cofinite topology]]. | * 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. | * 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 topolgical 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 image of a compact space under a continuous function is compact. |
Revision as of 07:14, 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 topolgical 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 is compact.