{{Short description|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.
==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 ''T''<sub>0</sub> 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''<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 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.
==See also==
==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