Editing Uniform space (section)

==Topology of uniform spaces==

Every uniform space <math>X</math> becomes a [[topological space]] by defining a nonempty subset <math>O \subseteq X</math> to be open if and only if for every <math>x \in O</math> there exists an entourage <math>V</math> such that <math>V[x]</math> is a subset of <math>O.</math> In this topology, the neighbourhood filter of a point <math>x</math> is <math>\{V[x] : V \in \Phi\}.</math> This can be proved with a recursive use of the existence of a "half-size" entourage. Compared to a general topological space the existence of the uniform structure makes possible the comparison of sizes of neighbourhoods: <math>V[x]</math> and <math>V[y]</math> are considered to be of the "same size".

The topology defined by a uniform structure is said to be '''{{visible anchor|induced by the uniformity}}'''. A uniform structure on a topological space is ''compatible'' with the topology if the topology defined by the uniform structure coincides with the original topology. In general several different uniform structures can be compatible with a given topology on <math>X.</math>

===Uniformizable spaces===
{{main|Uniformizable space}}
A topological space is called '''{{visible anchor|uniformizable}}''' if there is a uniform structure compatible with the topology.

Every uniformizable space is a [[completely regular space|completely regular]] topological space. Moreover, for a uniformizable space <math>X</math> the following are equivalent: 
* <math>X</math> is a [[Kolmogorov space]]
* <math>X</math> is a [[Hausdorff space]]
* <math>X</math> is a [[Tychonoff space]]
* for any compatible uniform structure, the intersection of all entourages is the diagonal <math>\{(x, x) : x \in X\}.</math>
Some authors (e.g. Engelking) add this last condition directly in the definition of a uniformizable space.

The topology of a uniformizable space is always a [[symmetric topology]]; that is, the space is an [[R0 space|R<sub>0</sub>-space]].

Conversely, each completely regular space is uniformizable. A uniformity compatible with the topology of a completely regular space <math>X</math> can be defined as the coarsest uniformity that makes all continuous real-valued functions on <math>X</math> uniformly continuous. A fundamental system of entourages for this uniformity is provided by all finite intersections of sets <math>(f \times f)^{-1}(V),</math> where <math>f</math> is a continuous real-valued function on <math>X</math> and <math>V</math> is an entourage of the uniform space <math>\mathbf{R}.</math> This uniformity defines a topology, which is clearly coarser than the original topology of <math>X;</math> that it is also finer than the original topology (hence coincides with it) is a simple consequence of complete regularity: for any <math>x \in X</math> and a neighbourhood <math>X</math> of <math>x,</math> there is a continuous real-valued function <math>f</math> with <math>f(x) = 0</math> and equal to 1 in the complement of <math>V.</math>

In particular, a compact Hausdorff space is uniformizable. In fact, for a compact Hausdorff space <math>X</math> the set of all neighbourhoods of the diagonal in <math>X \times X</math> form the ''unique'' uniformity compatible with the topology.

A Hausdorff uniform space is [[metrizable space|metrizable]] if its uniformity can be defined by a ''countable'' family of pseudometrics. Indeed, as discussed [[#Pseudometrics definition|above]], such a uniformity can be defined by a ''single'' pseudometric, which is necessarily a metric if the space is Hausdorff. In particular, if the topology of a [[vector space]] is Hausdorff and definable by a countable family of [[seminorm]]s, it is metrizable.