Editing Uniform space (section)

===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>