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A compact subset of a Hausdorff space X is closed. If X is not Hausdorff then a compact subset of X may fail to be a closed subset of X (see footnote for example). [b] If X is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example). [c] In any topological vector space (TVS), a compact subset is complete.
Lemma: A closed subset of a compact set is compact. Let K be a closed subset of a compact set T in R n and let C K be an open cover of K. Then U = R n \ K is an open set and = {} is an open cover of T. Since T is compact, then C T has a finite subcover ′, that also covers the smaller set K.
Every compact subset of a Hausdorff space is relatively compact. In a non-Hausdorff space, such as the particular point topology on an infinite set, the closure of a compact subset is not necessarily compact; said differently, a compact subset of a non-Hausdorff space is not necessarily relatively compact. Every compact subset of a (possibly ...
Functions with compact support on a topological space are those whose closed support is a compact subset of . If X {\displaystyle X} is the real line, or n {\displaystyle n} -dimensional Euclidean space, then a function has compact support if and only if it has bounded support , since a subset of R n {\displaystyle \mathbb {R} ^{n}} is compact ...
A tube in is a subset of the form where is an open subset of . It contains all the slices { x } × Y {\displaystyle \{x\}\times Y} for x ∈ U {\displaystyle x\in U} . Tube Lemma — Let X {\displaystyle X} and Y {\displaystyle Y} be topological spaces with Y {\displaystyle Y} compact, and consider the product space X × Y . {\displaystyle X ...
Precompact sets share a number of properties with compact sets. Like compact sets, a finite union of totally bounded sets is totally bounded. Unlike compact sets, every subset of a totally bounded set is again totally bounded. The continuous image of a compact set is compact. The uniformly continuous image of a precompact set is precompact.
Some examples of spaces that are not limit point compact: (1) The set of all real numbers with its usual topology, since the integers are an infinite set but do not have a limit point in ; (2) an infinite set with the discrete topology; (3) the countable complement topology on an uncountable set. Every countably compact space (and hence every ...
In a countably compact space, every locally finite family of nonempty subsets is finite. [10] [11] Every countably compact paracompact space is compact. [12] [11] More generally, every countably compact metacompact space is compact. [13] Every countably compact Hausdorff first-countable space is regular. [14] [15]