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# Hausdorff distance

Let $(X,d)$ be a metric space, then denote by $\mathcal{F}_X$ the family of all closed and bounded subsets of $X$. Given $A\in \mathcal{F}_X$, let denote by $N_r(A)$ the *neighborhood of $A$ of radius $r$*:
$$\cup_{x\in A} B(x,r).$$

Then the *Hausdorff distance* between $A,B\in \mathcal{F}_X$ is given by
$$\delta(A,B) = \max\{\inf \{r>0 : B\subset N_r(A)\},\inf \{r>0 : A\subset N_r(B)\}\}.$$

**Theorem.** $\delta$ is a metric.

**Theorem.** If $(X,d)$ is complete, then so is $(\mathcal{F}_X,\delta)$.