In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance. These maps are the morphisms in the category of metric spaces, Met.[1] Such functions are always continuous functions.
They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps.
Specifically, suppose that and are metric spaces and is a function from to . Thus we have a metric map when, for any points and in ,
Here and denote the metrics on and respectively.
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