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 on a set X is a filter on X × X such that (1) every U ∈ contains the diagonal {(x, x)}, (2) if U ∈ , its inverse is also in , and (3) if U ∈ , ∃ V ∈ such that V V ⊂ U.
$ (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, y ∈ X there is a cover C ∈ μ, no element of which contains both x and y.
* Examples: A p-adic structure; The additive uniformity on R defined by V_ε:= {(x, y) ∈ R × R | |x–y| < ε}, for ε > 0, and := {U | ∃ ε: V_ε ⊂ 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 ∈ .
* Uniform continuity: A function f : X → Y, with (X, ) and (Y, ') uniform spaces, is uniformly continuous if

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

Special Types and Generalizations
* H-equivalent pairs: Two uniformities and ' 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 is said to be H-singular if no distinct uniformity on X is H-equivalent to .
@ 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].

main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 11 feb 2012