In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

a uniform property or uniform invariant is a property of a

uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uni ...

which is invariant under uniform isomorphisms. Since uniform spaces come as

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

s and uniform isomorphisms are

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

s, every

topological property In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spa ...

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 is completely regular). * Complete. A uniform space ''X'' is complete if every Cauchy net in ''X'' converges (i.e. has a

limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...

in ''X''). * Totally bounded (or Precompact). A uniform space ''X'' is

totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “si ...

if for each entourage ''E'' ⊂ ''X'' × ''X'' there is a finite cover 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 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 is

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

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 In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest t ...

is constant. * Uniformly disconnected. A uniform space ''X'' is

uniformly disconnected In topology and related areas of mathematics a uniformly connected space or Cantor connected space is a uniform space ''U'' such that every uniformly continuous function from ''U'' to a discrete uniform space is constant. A uniform space ''U'' ...

if it is not uniformly connected.

References

* *{{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 Uniform spaces

Uniform properties

See also

References