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The set $𝒰{𝒟}_r$ of point sets of ${ℝ}^n,n\ge 1$, having the property that their minimal interpoint distance is greater than a given strictly positive constant $r\>0$ is shown to be equippable by a metric for which it is a compact topological space and such that the Hausdorff metric on the subset $𝒰{𝒟}_{r,f}\subset 𝒰{𝒟}_r$ of the finite point sets is compatible with the restriction of this topology to $𝒰{𝒟}_{r,f}$. We show that its subsets of Delone sets of given constants in ${ℝ}^n,n\ge 1$, are compact. Three (classes of) metrics, whose one of crystallographic nature,...