Compact space: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Richard Pinch
(→‎Examples: wrong ttheorem!)
imported>Richard Pinch
(added properties)
Line 25: Line 25:
* 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

This article is developing and not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

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