Compact space: Difference between revisions
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In [[mathematics]], a compact set is a [[set]] for which every covering of that set by a collection of | In [[mathematics]], a compact set is a [[set]] for which every covering of that set by a collection of [[open set]]s has a finite subcovering. If the set is a subset of a [[metric space]] then compactness is equivalent to the set being [[completeness|complete]] 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 [[normed space]], such as the Euclidean spaces, then compactness is equivalent to that set being closed and [[bounded set|bounded]]. | ||
==Cover and subcover of a set== | ==Cover and subcover of a set== |
Revision as of 06:30, 25 June 2008
In mathematics, a compact set is a set for which every covering of that set by a collection of open sets has a finite subcovering. If the set is a subset of a metric space then compactness is equivalent to the set being complete and 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 normed space, such as the Euclidean spaces, then compactness is equivalent to that set being closed and bounded.
Cover and subcover of a set
Let A be a subset of a set X. A cover for A is any collection of sets of the form , where is an arbitrary index set, such that . For any such cover , a set of the form with and such that is said to be a subcover of .
Formal definition of compact 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).