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Let $(X,d)$ be a complete metric space and consider \begin{align*} BC(X)&= \lbrace C\subset X\;|\;C\neq\emptyset\text {, closed and bounded} \rbrace\cr \mathrm{Fin}(X)&= \lbrace F\subset X\;|\;F\text{ is finite} \rbrace\subset BC(X)\cr \mathcal{K}(X)&= \lbrace K\subset X\;|\;K\text{ is compact} \rbrace\cr \end{align*} Consider the metric space $(BC(X), d_H)$ where $d_H$ is the Hausdorff distance (for the definition of $d_H$ see the Wikipedia entry on $d_H$)

I don't know how to prove that $\mathrm{Fin}(X)\subset BC(X)$ is dense in $\mathcal K(X)$ (with respect to $d_H$).

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    Correct, you need local total boundedness to show Fin is dense in BC...2012-03-24

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Suppose that $K\in\mathcal{K}(X)$ and $\epsilon>0$. Since $K$ is compact, it has a finite cover by $\epsilon$-balls, say $\{B(x_1,\epsilon),\dots,B(x_n,\epsilon)\}$. Let $F=\{x_1,x_2,\dots,x_n\}$. What can you say about $d_H(F,K)$? You may find it easiest to think of $d_H$ in terms of the characterization $d_H(X,Y)=\inf\{\epsilon>0:X\subseteq Y_\epsilon\text{ and }Y\subseteq X_\epsilon\}$ given in the Wikipedia article right after the definition.