We consider 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 Hausdorff radius of F and written r H ( F ), is bounded by the resolution of D . We call a closed set F 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 2 r . 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 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.