Uniform property

Updated on Nov 17, 2023

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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 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[x_i]. 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.

References

Uniform property Wikipedia

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