Uniformly continuous map: Difference between revisions

Revision as of 23:02, 24 November 2008

Definition

Between uniform spaces

Suppose $(X, U)$ and $(Y, V)$ are uniform spaces (in other words, $X$ and $Y$ are sets and $U$ and $V$ are uniform structures on $X$ and $Y$ respectively). A function $f : X \to Y$ is termed a uniformly continuous map if the following holds: For any $V \in V$ (i.e., for every entourage of $Y$ ) there exists a $U \in U$ such that $(a, b) \in U ⟹ (f (a), f (b)) \in V$ .

Between metric spaces

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:

$\forall ε > 0 \exists δ > 0 : d_{X} (a, b) < δ ⟹ d_{Y} (f (a), f (b)) < ε$ .

Revision as of 22:47, 24 November 2008 (view source) Vipul (talk \| contribs) (→‎Between metric spaces) ← Older edit		Revision as of 23:02, 24 November 2008 (view source) Vipul (talk \| contribs) (→‎Between metric spaces) Newer edit →
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	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:		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 : ~~\left\|~~ d_X(a,b) ~~\right\|~~ < \delta \implies ~~\left\|~~ d_Y(f(a),f(b)) ~~\right\|~~ < \varepsilon</math>.		<math>\forall \ \varepsilon > 0 \ \exists \delta > 0 : d_X(a,b) < \delta \implies d_Y(f(a),f(b)) < \varepsilon</math>.