Editing Uniform space (section)

===Entourage definition===

This definition adapts the presentation of a topological space in terms of [[neighborhood system]]s. A nonempty collection <math>\Phi</math> of subsets of <math>X \times X</math> is a '''{{visible anchor|uniform structure|Uniform structure}}''' (or a '''{{visible anchor|uniformity|Uniformity}}''') if it satisfies the following axioms:
# If <math>U\in\Phi</math> then <math>\Delta \subseteq U,</math> where <math>\Delta = \{(x,x) : x \in X\}</math> is the diagonal on <math>X \times X.</math>
# If <math>U\in\Phi</math> and <math>U \subseteq V \subseteq X \times X</math> then <math>V\in\Phi.</math>
# If <math>U\in\Phi</math> and <math>V\in\Phi</math> then <math>U \cap V \in \Phi.</math>
# If <math>U\in\Phi</math> then there is some <math>V \in\Phi</math> such that <math>V \circ V \subseteq U</math>, where <math>V \circ V</math> denotes the composite of <math>V</math> with itself. The [[Composition of relations|composite]] of two subsets <math>V</math> and <math>U</math> of <math>X \times X</math> is defined by <math display=block>V \circ U = \{(x,z) ~:~ \text{ there exists } y \in X \, \text{ such that } \, (x,y) \in U \wedge (y,z) \in V \,\}.</math>
# If <math>U\in\Phi</math> then <math>U^{-1} \in \Phi,</math> where <math>U^{-1} = \{(y,x) : (x,y)\in U\}</math> is the [[Converse relation|inverse]] of <math>U.</math>

The non-emptiness of <math>\Phi</math> taken together with (2) and (3) states that <math>\Phi</math> is a [[Filter (set theory)|filter]] on <math>X \times X.</math> If the last property is omitted we call the space '''{{visible anchor|quasiuniform}}'''.  An element <math>U</math> of <math>\Phi</math> is called a '''{{visible anchor|vicinity}}''' or '''{{visible anchor|entourage}}''' from the [[French language|French]] word for ''surroundings''.

One usually writes <math>U[x] = \{y : (x, y) \in U\} = \operatorname{pr}_2(U \cap (\{x\} \times X)\,),</math> where <math>U \cap (\{x\} \times X)</math> is the vertical cross section of <math>U</math> and <math>\operatorname{pr}_2</math> is the canonical projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the "<math>y = x</math>" diagonal; all the different <math>U[x]</math>'s form the vertical cross-sections. If <math>(x, y) \in U</math> then one says that <math>x</math> and <math>y</math> are '''{{visible anchor|U-closed|entourage-close|text=<math>U</math>-close}}'''. Similarly, if all pairs of points in a subset <math>A</math> of <math>X</math> are <math>U</math>-close (that is, if <math>A \times A</math> is contained in <math>U</math>), <math>A</math> is called ''<math>U</math>-small''. An entourage <math>U</math> is '''{{visible anchor|symmetric}}''' if <math>(x, y) \in U</math> precisely when <math>(y, x) \in U.</math> The first axiom states that each point is <math>U</math>-close to itself for each entourage <math>U.</math> The third axiom guarantees that being "both <math>U</math>-close and <math>V</math>-close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage <math>U</math> there is an entourage <math>V</math> that is "not more than half as large". Finally, the last axiom states that the property "closeness" with respect to a uniform structure is symmetric in <math>x</math> and <math>y.</math>

A '''{{visible anchor|base of entourages|base}}''' or '''{{visible anchor|fundamental system of entourages}}''' (or '''vicinities''') of a uniformity <math>\Phi</math> is any set <math>\mathcal{B}</math> of entourages of <math>\Phi</math> such that every entourage of <math>\Phi</math> contains a set belonging to <math>\mathcal{B}.</math> Thus, by property 2 above, a fundamental systems of entourages <math>\mathcal{B}</math> is enough to specify the uniformity <math>\Phi</math> unambiguously: <math>\Phi</math> is the set of subsets of <math>X \times X</math> that contain a set of <math>\mathcal{B}.</math> Every uniform space has a fundamental system of entourages consisting of symmetric entourages.

Intuition about uniformities is provided by the example of [[metric space]]s: if <math>(X, d)</math> is a metric space, the sets
<math display=block>U_a = \{(x,y) \in X \times X : d(x,y) \leq a\} \quad \text{where} \quad a > 0</math>
form a fundamental system of entourages for the standard uniform structure of <math>X.</math> Then <math>x</math> and <math>y</math> are <math>U_a</math>-close precisely when the distance between <math>x</math> and <math>y</math> is at most <math>a.</math>

A uniformity <math>\Phi</math> is ''finer'' than another uniformity <math>\Psi</math> on the same set if <math>\Phi \supseteq \Psi;</math> in that case <math>\Psi</math> is said to be ''coarser'' than <math>\Phi.</math>