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

Consider the metric space ${\displaystyle [0,1/2]}$ with the Euclidean metric. Then the function ${\displaystyle f(x)=x^{2}}$ is a metric map, since for ${\displaystyle x\neq y}$, ${\displaystyle |f(x)-f(y)|=|x+y||x-y|<|x-y|}$.

The function composition of two metric maps is another metric map, and the identity map ${\displaystyle \mathrm {id} _{M}\colon M\rightarrow M}$ on a metric space ${\displaystyle M}$ 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.

One can say that ${\displaystyle f}$ is strictly metric if the inequality is strict for every two different points. Thus a contraction mapping is strictly metric, but not necessarily the other way around. Note that an isometry is never strictly metric, except in the degenerate case of the empty space or a single-point space.

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

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