Editing Uniform property (section)

==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.