[[Category of metric spaces]]
# Category of metric spaces and isometries

The category $\cat{IsoMet}$ contains [[Metric space|metric spaces]] as its objects and isometries as its morphisms, #m/def/anal 
i.e. if $(X, d_{X})$ and $(Y, d_{Y})$ are metric spaces,
a morphism $f \in \cat{IsoMet}((X,d_{X}), (Y,d_{Y}))$ is a function $f:X\to Y$ such that
$$
\begin{align*}
d_{X}(a,b) = d_{Y}(f(a),f(b))
\end{align*}
$$
for any $a, b \in X$.


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