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Finite approximations to the Kuratowski/Fréchet embedding

Let $(X,d)$ be a compact doubling metric space with doubling constant $C>0$. Let $\{\mathbb{X}_n\}_{n=0}^{\infty}$ be a sequences of finite subsets of $X$ with $$ \left\{B\left(x_k,\frac1{n}\right)...
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