Editing Uniform property

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

Since uniform spaces come as [[topological space]]s and uniform isomorphisms are [[homeomorphism]]s, 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 space|separated]] if the intersection of all [[entourage (topology)|entourage]]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 space|Hausdorff]] (or simply [[T0 space|''T''<sub>0</sub>]] since every uniform space is [[completely regular]]).
* '''Complete'''. A uniform space ''X'' is [[complete space|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 (topology)|cover]] {''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 space|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 space|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.

==See also==

*[[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]]