Uniform property
Object of study in the category of uniform topological spaces
In the mathematical field of topology a uniform property or uniform invariant is a property of a uniform space that 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.
01Uniform 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 {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[xi]. 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.
Sources and credits
This article is adapted from the Wikipedia article “Uniform property”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
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