Uniformly continuous map of metric spaces: Difference between revisions

Latest revision as of 23:33, 24 November 2008

Definition

Definition in terms of the metric

Suppose $(X,d_{X})$ and $(Y,d_{Y})$ are metric spaces (in other words, $X$ and $Y$ are sets and $d_{X}$ and $d_{Y}$ are metrics on $X$ and $Y$ respectively). A function $f:X\to Y$ is termed a uniformly continuous map if the following holds:

$\forall \ \varepsilon >0\ \exists \delta >0:d_{X}(a,b)<\delta \implies d_{Y}(f(a),f(b))<\varepsilon$ .

Definition in terms of the uniform structure

Suppose $(X,d_{X})$ and $(Y,d_{Y})$ are metric spaces. A map $f:X\to Y$ is termed uniformly continuous if $f$ is a uniformly continuous map from $X$ to $Y$ with respect to the induced uniform structures on $X$ and $Y$ from their respective metrics.

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 ==Definition==
-===Definition with symbols===
+===Definition in terms of the metric===
 Suppose <math>(X,d_X)</math> and <math>(Y,d_Y)</math> are [[metric space]]s (in other words, <math>X</math> and <math>Y</math> are sets and <math>d_X</math> and <math>d_Y</math> are metrics on <math>X</math> and <math>Y</math> respectively). A function <math>f:X \to Y</math> is termed a '''uniformly continuous map''' if the following holds:
 <math>\forall \ \varepsilon > 0 \ \exists \delta > 0 : d_X(a,b) < \delta \implies d_Y(f(a),f(b)) < \varepsilon</math>.
+===Definition in terms of the uniform structure===
+Suppose <math>(X,d_X)</math> and <math>(Y,d_Y)</math> are [[metric space]]s. A map <math>f:X \to Y</math> is termed uniformly continuous if <math>f</math> is a [[defining ingredient::uniformly continuous map]] from <math>X</math> to <math>Y</math> with respect to the [[metric induces uniform structure|induced uniform structures]] on <math>X</math> and <math>Y</math> from their respective metrics.