Editing Metric map (section)

{{Short description|Function between metric spaces that does not increase any distance}}
In the [[mathematics|mathematical]] theory of [[metric space]]s, a '''metric map''' is a [[Function (mathematics)|function]] between metric spaces that does not increase any distance. These maps are the [[morphism]]s in the [[category of metric spaces]], '''Met'''.{{r|isbell}} Such functions are always  [[continuous function]]s.
They are also called [[Lipschitz continuity|Lipschitz functions]] with [[Lipschitz constant]] 1, '''nonexpansive maps''', '''nonexpanding maps''', '''weak contractions''', or '''short maps'''.

Specifically, suppose that <math>X</math> and <math>Y</math> are metric spaces and <math>f</math> is a [[function (mathematics)|function]] from <math>X</math> to <math>Y</math>. Thus we have a metric map when, [[for any]] points <math>x</math> and <math>y</math> in <math>X</math>,
<math display=block> d_{Y}(f(x),f(y)) \leq d_{X}(x,y) . \! </math>
Here <math>d_X</math> and <math>d_Y</math> denote the metrics on <math>X</math> and <math>Y</math> respectively.