Metric map

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 X {\displaystyle X} and Y {\displaystyle Y} are metric spaces and f {\displaystyle f} is a function from X {\displaystyle X} to Y {\displaystyle Y} . Thus we have a metric map when, for any points x {\displaystyle x} and y {\displaystyle y} in X {\displaystyle X} , d Y ( f ( x ) , f ( y ) ) ≤ d X ( x , y ) . {\displaystyle d_{Y}(f(x),f(y))\leq d_{X}(x,y).\!} Here d X {\displaystyle d_{X}} and d Y {\displaystyle d_{Y}} denote the metrics on X {\displaystyle X} and Y {\displaystyle Y} respectively.

Consider the metric space [ 0 , 1 / 2 ] {\displaystyle [0,1/2]} with the Euclidean metric. Then the function f ( x ) = x 2 {\displaystyle f(x)=x^{2}} is a metric map, since for x ≠ y {\displaystyle x\neq y} , | f ( x ) − f ( y ) | = | x + y | | x − y | < | x − y | {\displaystyle |f(x)-f(y)|=|x+y||x-y|<|x-y|} . In this example the Lipschitz constant is 1, that implies a metric map.

The function composition of two metric maps is another metric map, and the identity map i d M : M → M {\displaystyle \mathrm {id} _{M}\colon M\rightarrow M} on a metric space M {\displaystyle M} is a metric map, which is also the identity element for function composition. Thus metric spaces together with metric maps form a category Met. Met is a subcategory of the category of metric spaces and Lipschitz functions. A map between metric spaces is an isometry if and only if it is a bijective metric map whose inverse is also a metric map. Thus the isomorphisms in Met are precisely the isometries.

A mapping T : X → N ( X ) {\displaystyle T\colon X\to {\mathcal {N}}(X)} from a metric space X {\displaystyle X} to the family of nonempty subsets of X {\displaystyle X} is said to be Lipschitz if there exists L ≥ 0 {\displaystyle L\geq 0} such that H ( T x , T y ) ≤ L d ( x , y ) , {\displaystyle H(Tx,Ty)\leq Ld(x,y),} for all x , y ∈ X {\displaystyle x,y\in X} , where H {\displaystyle H} is the Hausdorff distance. When L = 1 {\displaystyle L=1} , T {\displaystyle T} is called nonexpansive, and when L < 1 {\displaystyle L<1} , T {\displaystyle T} is called a contraction.

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