Uniform property

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title: "Uniform property" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["uniform-spaces"] description: "Object of study in the category of uniform topological spaces" topic_path: "general/uniform-spaces" source: "https://en.wikipedia.org/wiki/Uniform_property" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Object of study in the category of uniform topological spaces ::

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

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::