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 ...
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 un ...
which is
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iterat ...
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 poi ...
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 isomo ...
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 spaces ...
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
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
).
* Complete. A uniform space ''X'' is
complete if every
Cauchy net
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codomai ...
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 cont ...
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 “size ...
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 Briti ...
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
*
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 spaces ...
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