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).
Totally bounded (or Precompact). A uniform space X is
totally bounded if for each entourage E ⊂ X × 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 Exi]. 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).
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).
Totally bounded (or Precompact). A uniform space X is
totally bounded if for each entourage E ⊂ X × 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 Exi]. 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).