Editing Glossary of general topology (section)

== U ==

;Ultra-connected: A space is ultra-connected if no two non-empty closed sets are disjoint.<ref name="ss29"/> Every ultra-connected space is path-connected.

;[[Ultrametric space|Ultrametric]]: A metric is an ultrametric if it satisfies the following stronger version of the [[triangle inequality]]: for all ''x'', ''y'', ''z'' in ''M'', ''d''(''x'', ''z'') ≤ max(''d''(''x'', ''y''), ''d''(''y'', ''z'')).

;[[Uniform isomorphism]]: If ''X'' and ''Y'' are [[uniform space]]s, a uniform isomorphism from ''X'' to ''Y'' is a bijective function ''f'' : ''X'' → ''Y'' such that ''f'' and ''f''<sup>−1</sup> are [[uniformly continuous]]. The spaces are then said to be uniformly isomorphic and share the same [[uniform properties]].

;[[Uniformizable]]/Uniformisable: A space is uniformizable if it is homeomorphic to a uniform space.

;[[Uniform space]]: A [[uniform space]] is a set ''X'' equipped with a nonempty collection Φ of subsets of the [[Cartesian product]] ''X'' × ''X'' satisfying the following [[axiom]]s:

:# if ''U'' is in Φ, then ''U'' contains { (''x'', ''x'') | ''x'' in ''X'' }.
:# if ''U'' is in Φ, then { (''y'', ''x'') | (''x'', ''y'') in ''U'' } is also in Φ
:# if ''U'' is in Φ and ''V'' is a subset of ''X'' × ''X'' which contains ''U'', then ''V'' is in Φ
:# if ''U'' and ''V'' are in Φ, then ''U'' ∩ ''V'' is in Φ
:# if ''U'' is in Φ, then there exists ''V'' in Φ such that, whenever (''x'', ''y'') and (''y'', ''z'') are in ''V'', then (''x'', ''z'') is in ''U''.

:The elements of Φ are called '''entourages''', and Φ itself is called a '''uniform structure''' on ''X''. The uniform structure induces a topology on ''X'' where the basic neighborhoods of ''x'' are sets of the form {''y'' : (''x'',''y'')∈''U''} for ''U''∈Φ.

;Uniform structure: See '''[[Uniform space]]'''.