In the
mathematical field of
topology a uniform property or uniform invariant is a property of a
uniform space 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 iteratio ...
under
uniform isomorphism In the mathematical field of topology a uniform isomorphism or is a special isomorphism between uniform spaces that respects uniform properties. Uniform spaces with uniform maps form a category. An isomorphism between uniform spaces is called a un ...
s.
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''0 since every uniform space is
completely regular).
* Complete. A uniform space ''X'' is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
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 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 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 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
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