Uniform uniformity

The uniform uniform continuity connects the terms uniform and uniform continuity .

Be , metric spaces, is a subset limited , continuous functions . The family of functions is called uniformly uniformly continuous if: ${\displaystyle (X,d_{X})}$ ${\displaystyle (Y,d_{Y})}$ ${\displaystyle F\subset C_{b}(X,Y)}$ ${\displaystyle F}$

For all one exists , such that for all and for all true: ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta >0}$ ${\displaystyle x,x'\in X}$ ${\displaystyle f\in F}$

${\displaystyle d_{X}(x,x')\leq \delta \Rightarrow d_{Y}\left(f(x),f(x')\right)\leq \varepsilon }$.

That means, if one gives one, one finds one , so that the statement applies to all functions of the family and to all points of the room. So depends only on, neither on nor on . ${\displaystyle \varepsilon }$ ${\displaystyle \delta }$ ${\displaystyle \delta }$ ${\displaystyle \varepsilon }$ ${\displaystyle f}$ ${\displaystyle x}$

Individual evidence

1. ^ Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture , Bibliographisches Institut Mannheim (1978), ISBN 3-411-00121-6 , paragraph 5.8, exercise 41