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].

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].