Uniform property
Encyclopedia
In the mathematical
field of topology
a uniform property or uniform invariant is a property of a uniform space
which is invariant
under uniform isomorphism
s.
Since uniform spaces come as topological space
s and uniform isomorphisms are homeomorphism
s, 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
a uniform property or uniform invariant is a property of a uniform space
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 which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
which is invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
under uniform isomorphism
Uniform isomorphism
In the mathematical field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces which respects uniform properties.-Definition:...
s.
Since uniform spaces come as topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s and uniform isomorphisms are homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
s, every topological property
Topological property
In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic...
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 HausdorffHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
(or simply T0 since every uniform space is completely regular). - Complete. A uniform space X is completeComplete spaceIn mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
if every Cauchy netCauchy netIn mathematics, a Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.A net is a Cauchy net if for every entourage V there exists γ such that for all α, β ≥ γ, is a member of V. More generally, in a Cauchy space, a net is Cauchy if the filter generated by the...
in X converges (i.e. has a limit pointLimit pointIn mathematics, a limit point of a set S in a topological space X is a point x in 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 contains a point of S other than x itself. Note that x does not have to be an element of S...
in X). - Totally bounded (or Precompact). A uniform space X is totally bounded if for each entourage E ⊂ X × X there is a finite coverCover (topology)In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset. Formally, ifC = \lbrace U_\alpha: \alpha \in A\rbrace...
{Ui} of X such that Ui × Ui is contained in E for all i. Equivalently, X is totally bounded if for each entourage E there exists a finite subset {xi} of X such that X is the union of all E[xi]. In terms of uniform covers, X is totally bounded if every uniform cover has a finite subcover. - Compact. A uniform space is compactCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
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.