# Uniform property

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{{Short description|Object of study in the category of uniform topological spaces}}In the [mathematical](/source/mathematics) field of [topology](/source/topology) a '''uniform property''' or '''uniform invariant''' is a property of a [uniform space](/source/uniform_space) that is [invariant](/source/invariant_(mathematics)) under [uniform isomorphism](/source/uniform_isomorphism)s.

Since uniform spaces come as [topological space](/source/topological_space)s and uniform isomorphisms are [homeomorphism](/source/homeomorphism)s, every [topological property](/source/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](/source/separated_space) if the intersection of all [entourage](/source/entourage_(topology))s 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](/source/Hausdorff_space) (or simply [''T''<sub>0</sub>](/source/T0_space) since every uniform space is [completely regular](/source/completely_regular)).
* '''Complete'''. A uniform space ''X'' is [complete](/source/complete_space) if every [Cauchy net](/source/Cauchy_net) in ''X'' converges (i.e. has a [limit point](/source/limit_point) in ''X'').
* '''Totally bounded''' (or '''Precompact'''). A uniform space ''X'' is [totally bounded](/source/totally_bounded) if for each entourage ''E'' ⊂ ''X'' × ''X'' there is a finite [cover](/source/cover_(topology)) {''U''<sub>''i''</sub>} of ''X'' such that ''U''<sub>''i''</sub> × ''U''<sub>''i''</sub> is contained in ''E'' for all ''i''. Equivalently, ''X'' is totally bounded if for each entourage ''E'' there exists a finite subset {''x''<sub>''i''</sub>} of ''X'' such that ''X'' is the union of all ''E''[''x''<sub>''i''</sub>]. In terms of uniform covers, ''X'' is totally bounded if every uniform cover has a finite subcover.
* '''Compact'''. A uniform space is [compact](/source/compact_space) 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](/source/Uniformly_connected_space) if every [uniformly continuous function](/source/uniformly_continuous_function) from ''X'' to a [discrete uniform space](/source/discrete_uniform_space) is constant.
* '''Uniformly disconnected'''. A uniform space ''X'' is [uniformly disconnected](/source/uniformly_disconnected) if it is not uniformly connected.

==See also==

*[Topological property](/source/Topological_property)

==References==

*{{cite book | last = James | first = I. M. | title = Introduction to Uniform Spaces | url = https://archive.org/details/introductiontoun0000jame | url-access = registration | publisher = Cambridge University Press | location = Cambridge, UK | year = 1990 | isbn = 0-521-38620-9}}
*{{cite book | last = Willard | first = Stephen | title = General Topology | url = https://archive.org/details/generaltopology00will_0 | url-access = registration | publisher = Addison-Wesley | location = Reading, Massachusetts | year = 1970 | isbn = 0-486-43479-6 }}

Category:Uniform spaces

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Adapted from the Wikipedia article [Uniform property](https://en.wikipedia.org/wiki/Uniform_property) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Uniform_property?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.