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 E ⊂ X × X there is a finite cover {U_i} 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 {x_i} 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.