Editing Uniform space (section)

==Definition==
There are three equivalent definitions for a uniform space.  They all consist of a space equipped with a uniform structure.

===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>

===Pseudometrics definition===

Uniform spaces may be defined alternatively and equivalently using systems of [[Pseudometric space|pseudometrics]], an approach that is particularly useful in [[functional analysis]] (with pseudometrics provided by [[seminorm]]s). More precisely, let <math>f : X \times X \to \R</math> be a pseudometric on a set <math>X.</math> The inverse images <math>U_a = f^{-1}([0, a])</math> for <math>a > 0</math> can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by the <math>U_a</math> is the uniformity defined by the single pseudometric <math>f.</math> Certain authors call spaces the topology of which is defined in terms of pseudometrics ''gauge spaces''.

For a ''family'' <math>\left(f_i\right)</math> of pseudometrics on <math>X,</math> the uniform structure defined by the family is the ''least upper bound'' of the uniform structures defined by the individual pseudometrics <math>f_i.</math> A fundamental system of entourages of this uniformity is provided by the set of ''finite'' intersections of entourages of the uniformities defined by the individual pseudometrics <math>f_i.</math> If the family of pseudometrics is ''finite'', it can be seen that the same uniform structure is  defined by a ''single'' pseudometric, namely the [[upper envelope]] <math>\sup_{} f_i</math> of the family.

Less trivially, it can be shown that a uniform structure that admits a [[countable]] fundamental system of entourages (hence in particular a uniformity defined by a countable family of pseudometrics) can be defined by a single pseudometric. A consequence is that ''any'' uniform structure can be defined as above by a (possibly uncountable) family of pseudometrics (see Bourbaki: General Topology Chapter IX §1 no. 4).

===Uniform cover definition===

A '''uniform space''' <math>(X, \Theta)</math> is a set <math>X</math> equipped with a distinguished family of coverings <math>\Theta,</math> called "uniform covers", drawn from the set of [[Cover (topology)|coverings]] of <math>X,</math> that form a [[Filter (mathematics)#General definition: Filter on a partially ordered set|filter]] when ordered by star refinement. One says that a cover <math>\mathbf{P}</math> is a ''[[star refinement]]'' of cover <math>\mathbf{Q},</math> written <math>\mathbf{P} <^* \mathbf{Q},</math> if for every <math>A \in \mathbf{P},</math> there is a <math>U \in \mathbf{Q}</math> such that if <math>A \cap B \neq \varnothing,B \in \mathbf{P},</math> then <math>B \subseteq U.</math> Axiomatically, the condition of being a filter reduces to:

# <math>\{X\}</math> is a uniform cover (that is, <math>\{X\} \in \Theta</math>).
# If <math>\mathbf{P} <^* \mathbf{Q}</math> with <math>\mathbf{P}</math> a uniform cover and <math>\mathbf{Q}</math> a cover of <math>X,</math> then <math>\mathbf{Q}</math> is also a uniform cover.
# If <math>\mathbf{P}</math> and <math>\mathbf{Q}</math> are uniform covers then there is a uniform cover <math>\mathbf{R}</math> that star-refines both <math>\mathbf{P}</math> and <math>\mathbf{Q}</math>

Given a point <math>x</math> and a uniform cover <math>\mathbf{P},</math> one can consider the union of the members of <math>\mathbf{P}</math> that contain <math>x</math> as a typical neighbourhood of <math>x</math> of "size" <math>\mathbf{P},</math> and this intuitive measure applies uniformly over the space.

Given a uniform space in the entourage sense, define a cover <math>\mathbf{P}</math> to be uniform if there is some entourage <math>U</math> such that for each <math>x \in X,</math> there is an <math>A \in \mathbf{P}</math> such that <math>U[x] \subseteq A.</math> These uniform covers form a uniform space as in the second definition. Conversely, given a uniform space in the uniform cover sense, the supersets of <math>\bigcup \{A \times A : A \in \mathbf{P}\},</math> as <math>\mathbf{P}</math> ranges over the uniform covers, are the entourages for a uniform space as in the first definition. Moreover, these two transformations are inverses of each other. <ref name="isarmathlib_UniformSpace_ZF_2">{{cite web |url=https://isarmathlib.org/UniformSpace_ZF_2.html |title=IsarMathLib.org |accessdate=2021-10-02 }}</ref>