Uniform property


Uniform property

In the mathematical field of topology a uniform property or uniform invariant is a property of a uniform space which is invariant under uniform isomorphisms.

Since uniform spaces come are 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 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 {"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 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 is constant.
* Uniformly disconnected. A uniform space "X" is uniformly 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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