Uniformities / Uniform Spaces| 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 \(\circ\) 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
\(\mathbb R\) defined by Vε:=
{(x, y) ∈ \(\mathbb R\) × \(\mathbb R\) |
|x−y| < 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 : X → Y, 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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send feedback and suggestions to bombelli at olemiss.edu – modified 26 jan 2016