In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
field of
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
a uniform property or uniform invariant is a property of a
uniform space
In the mathematical field of topology, a uniform space is a topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform con ...
that is
invariant under
uniform isomorphisms.
Since uniform spaces come as
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s and uniform isomorphisms are
homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
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 space ...
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 is any completely regular space that is also a ...
).
* 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 function whose domain is a directed set. The codomain of this function is usually some topological space. Nets directly generalize ...
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 contains a point of S other than x itself. A ...
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, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
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
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
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 to ...
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'' is ...
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 space ...
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