Correspondence between Topological and Discrete Connectivities in Hausdorff Discretization

We consider \emph{Hausdorff discretization} from a metric space $E$ to a discrete subspace $D$, which associates to a closed subset $F$ of $E$ any subset $S$ of $D$ minimizing the Hausdorff distance between $F$ and $S$; this minimum distance, called the \emph{Hausdorff radius} of $F$ and written $r_H(F)$, is bounded by the resolution of $D$. We call a closed set $F$ \emph{separated} if it can be partitioned into two non-empty closed subsets $F_1$ and $F_2$ whose mutual distances have a strictly positive lower bound. Assuming some minimal topological properties of $E$ and $D$ (satisfied in $\R^n$ and $\Z^n$), we show that given a non-separated closed subset $F$ of $E$, for any $r>r_H(F)$, every Hausdorff discretization of $F$ is connected for the graph with edges linking pairs of points of $D$ at distance at most $2r$. When $F$ is connected, this holds for $r=r_H(F)$, and its greatest Hausdorff discretization belongs to the partial connection generated by the traces on $D$ of the balls of radius $r_H(F)$. However, when the closed set $F$ is separated, the Hausdorff discretizations are disconnected whenever the resolution of $D$ is small enough. In the particular case where $E=\R^n$ and $D=\Z^n$ with norm-based distances, we generalize our previous results for $n=2$. For a norm invariant under changes of signs of coordinates, the greatest Hausdorff discretization of a connected closed set is axially connected. For the so-called \emph{coordinate-homogeneous} norms, which include the $L_p$ norms, we give an adjacency graph for which all Hausdorff discretizations of a connected closed set are connected