Uniformities / Uniform Spaces  

In General
* Idea: A richer structure than a topology, due to A Weil (1937, entourage version) and Tukey (covering version).
$ (Entourages) A uniformity \(\cal U\) on a set X is a filter on X × X such that (1) every U ∈ \(\cal U\) contains the diagonal {(x, x)}, (2) if U ∈ \(\cal U\), its inverse is also in \(\cal U\), and (3) if U ∈ \(\cal U\), ∃ V ∈ \(\cal U\) such that V \(\circ\) VU.
$ (Coverings) A (separated) uniformity μ on X is a family of coverings of X which is a filter with respect to star-refinement (pre-uniformity), and such that for all x, yX there is a cover Cμ, no element of which contains both x and y.
* Examples: A p-adic structure; The additive uniformity on \(\mathbb R\) defined by Vε:= {(x, y) \(\mathbb R\) × \(\mathbb R\) | |xy| < ε}, for ε > 0, and \(\cal U\):= {U | ∃ ε: Vε ⊂ U}.
@ References: in Kelley 55; in Bourbaki 61; Isbell 64; in Pervin 64; in Schubert 68; Page 78; James 87; in Preuss 02; Künzi T&A(07) [survey]; Bridges & Vîţă 11 [using constructive logic].
> Online resources: see Wikipedia page.

Related Concepts > s.a. Approach Space; proximity; Uniform Cover; Uniform Equivalence.
* And other structure: A topological space is uniformizable iff it is completely regular (also, a gage space); Viceversa, the topology defined by a uniformity is always completely regular; A uniformity defines a proximity by A δ B iff (A × B) ∩ U ≠ Ø, for all U ∈ \(\cal U\).
* Uniform continuity: A function f : XY, with (X, \(\cal U\)) and (Y, \({\cal U}'\)) uniform spaces, is uniformly continuous if

for all V ∈ \({\cal U}'\), ∃ U ∈ \(\cal U\)    such that    (x, y) ∈ U  implies  (f(x), f(y)) ∈ V .

@ References: Banakh & Repovš T&A(10) [direct limits].

Special Types and Generalizations
* H-equivalent pairs: Two uniformities \(\cal U\) and \({\cal U}'\) on a set X are said to be H-equivalent if their corresponding Hausdorff uniformities on the set of all non-empty subsets of X induce the same topology; The uniformity \(\cal U\) is said to be H-singular if no distinct uniformity on X is H-equivalent to \(\cal U\).
@ Types of uniformities: Bouziad T&A(09) [\(H\)-equivalent pairs and H-singular uniformities].
@ Quasi-uniformities: de Jager & Künzi T&A(06) [atoms]; Özçag & Brown T&A(06) [and uniformities, textural view].

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