Editing Uniform continuity (section)

== Generalization to uniform spaces ==
Just as the most natural and general setting for continuity is [[topological space]]s, the most natural and general setting for the study of ''uniform'' continuity are the [[uniform space]]s. A function <math>f:X \to Y</math> between uniform spaces is called ''uniformly continuous'' if for every [[entourage (topology)|entourage]] ''<math>V</math>'' in ''<math>Y</math>'' there exists an entourage ''<math>U</math>'' in ''<math>X</math>'' such that for every <math>(x_1,x_2)</math> in <math>U</math> we have <math>(f(x_1),f(x_2))</math> in <math>V</math>.

In this setting, it is also true that uniformly continuous maps transform Cauchy sequences into Cauchy sequences.

Each [[compact Hausdorff space]] possesses exactly one uniform structure compatible with the topology. A consequence is a generalization of the Heine-Cantor theorem: each continuous function from a compact Hausdorff space to a uniform space is uniformly continuous.