Metric map

Template:Short description 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.Template:R 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 $X$ and $Y$ are metric spaces and $f$ is a function from $X$ to $Y$. Thus we have a metric map when, for any points $x$ and $y$ in $X$, $$d_Y(f(x),f(y))\le d_X(x,y).$$ Here $d_X$ and $d_Y$ denote the metrics on $X$ and $Y$ respectively.

Examples

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

Category of metric maps

The function composition of two metric maps is another metric map, and the identity map ${\mathrm{i}\mathrm{d}}_M:M\to M$ on a metric space $M$ 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.

Multivalued version

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

See also

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