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In the mathematical field of topology a uniform property or uniform invariant is a property of a uniform space which is invariant under uniform isomorphisms. Since uniform spaces come as topological spaces and uniform isomorphisms are homeomorphisms, every topological property of a uniform space is also a uniform property. This article is (mostly) concerned with uniform properties that are ''not'' topological properties. ==Uniform properties== * Separated. A uniform space ''X'' is separated if the intersection of all entourages is equal to the diagonal in ''X'' × ''X''. This is actually just a topological property, and equivalent to the condition that the underlying topological space is Hausdorff (or simply ''T''0 since every uniform space is completely regular). * Complete. A uniform space ''X'' is complete if every Cauchy net in ''X'' converges (i.e. has a limit point in ''X''). * Totally bounded (or Precompact). A uniform space ''X'' is totally bounded if for each entourage ''E'' ⊂ ''X'' × ''X'' there is a finite cover of ''X'' such that ''U''''i'' × ''U''''i'' is contained in ''E'' for all ''i''. Equivalently, ''X'' is totally bounded if for each entourage ''E'' there exists a finite subset of ''X'' such that ''X'' is the union of all ''E''(). In terms of uniform covers, ''X'' is totally bounded if every uniform cover has a finite subcover. * Compact. A uniform space is compact if it is complete and totally bounded. Despite the definition given here, compactness is a topological property and so admits a purely topological description (every open cover has a finite subcover). * Uniformly connected. A uniform space ''X'' is uniformly connected if every uniformly continuous function from ''X'' to a discrete uniform space is constant. * Uniformly disconnected. A uniform space ''X'' is uniformly disconnected if it is not uniformly connected. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Uniform property」の詳細全文を読む スポンサード リンク
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