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I started to read a bit Gromov's paper 'Metric Structures for Riemannian and Non-riemannian Spaces'. Tthe Hausdorf distance of two metric spaces $X$ and $Y$ is defined by using isometric embeddings $f:X\rightarrow Z$ and $g:Y\rightarrow Z$ into a third metric space $Z$ and then taking some infimum over all such maps $f,g$ and spaces $Z$. So I am wondering why/if such a space $Z$ and maps $f,g$ always exist?Maybe one can construct always a metric on the product space to obtain such a $Z$...

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For two metric spaces $(X_1,d_1)$ and $(X_2,d_2)$, define the distance $d$ on $X:=X_1 \times X_2$ by $d : ((x_1,x_2),(y_1,y_2)) \mapsto \max(d_1(x_1,y_1),d_2(x_2,y_2))$. Then $\phi_1 : x \mapsto (x,0)$ and $\phi_2 : x \mapsto (0,x)$ are embeddings of $X_1$ and $X_2$ respectively into $X$.

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