- Uniform property
In the mathematical field of

topology a**uniform property**or**uniform invariant**is a property of auniform space which is invariant underuniform isomorphism s.Since uniform spaces come are

topological space s and uniform isomorphisms arehomeomorphism s, everytopological 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"_{0}since every uniform space iscompletely regular ).

***Complete**. A uniform space "X" is complete if everyCauchy net in "X" converges (i.e. has alimit point in "X").

***Totally bounded**(or**Precompact**). A uniform space "X" istotally 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 iscompact 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" isuniformly connected if everyuniformly continuous function from "X" to adiscrete uniform space is constant.

***Uniformly disconnected**. A uniform space "X" isuniformly disconnected if every uniformly continuous function from a discrete uniform space to "X" is constant.**ee also***

Topological property **References***cite book | last = James | first = I. M. | title = Introduction to Uniform Spaces | publisher = Cambridge University Press | location = Cambridge, UK | year = 1990 | isbn = 0-521-38620-9

*cite book | last = Willard | first = Stephen | title = General Topology | publisher = Addison-Wesley | location = Reading, Massachusetts | year = 1970 | id = ISBN 0-486-43479-6 (Dover edition)

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