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 *V*_{ε}:=
{(*x*,* y*) **∈ **\(\mathbb R\) × \(\mathbb R\) |
|*x*−*y*| < *ε*}, for *ε* > 0,
and \(\cal U\):= {*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].

> __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

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