Uniform continuity

A property of a function (mapping) $ f: X \rightarrow Y $, where $ X $ and $ Y $ are metric spaces. It requires that for any $ \epsilon > 0 $ there is a $ \delta > 0 $ such that for all $ x _ {1} , x _ {2} \in X $ satisfying $ \rho ( x _ {1} , x _ {2} ) < \delta $, the inequality $ \rho ( f ( x _ {1} ), f ( x _ {2} )) < \epsilon $ holds.

If a mapping $ f: X \rightarrow Y $ is continuous on $ X $ and $ X $ is a compactum, then $ f $ is uniformly continuous on $ X $. The composite of uniformly-continuous mappings is uniformly continuous.

Uniform continuity of mappings occurs also in the theory of topological groups. For example, a mapping $ f: X _ {0} \rightarrow Y $, where $ X _ {0} \subset X $, $ X $ and $ Y $ topological groups, is said to be uniformly continuous if for any neighbourhood of the identity $ U _ {y} $ in $ Y $, there is a neighbourhood of the identity $ U _ {x} $ in $ X $ such that for any $ x _ {1} , x _ {2} \in X _ {0} $ satisfying $ x _ {1} x _ {2} ^ {-} 1 \in U _ {x} $( respectively, $ x _ {1} ^ {-} 1 x _ {2} \in U _ {x} $), the inclusion $ f ( x _ {1} ) [ f ( x _ {2} )] ^ {-} 1 \in U _ {y} $( respectively, $ [ f ( x _ {1} )] ^ {-} 1 f ( x _ {2} ) \in U _ {y} $) holds.

The notion of uniform continuity has been generalized to mappings of uniform spaces (cf. Uniform space).

References

[1]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[2]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[3]	J.L. Kelley, "General topology" , Springer (1975)
[4]	N. Bourbaki, "General topology" , Elements of mathematics , Springer (1989) (Translated from French)

Comments

There are several natural uniform structures on a topological group; the (confusing) statement above about uniform continuity of mappings between them can be interpreted in various ways.

References

[a1]	W. Roelcke, S. Dierolf, "Uniform structures on topological groups and their quotients" , McGraw-Hill (1981)
[a2]	R. Engelking, "General topology" , Heldermann (1989)