Uniform Property - Uniform 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 T0 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 EX × X there is a finite cover {Ui} of X such that Ui × Ui is contained in E for all i. Equivalently, X is totally bounded if for each entourage E there exists a finite subset {xi} 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.

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