Optimal conditions for connectedness of discretized sets

An offset discretization of a set $$X \subset {\mathbb {R}}^n$$ is obtained by taking the integer points inside closed neighborhood X certain radius. In this work we determine minimum threshold for radius, beyond which an arbitrary (possibly disconnected) always connected. The results hold broad class disconnected subsets $${\mathbb and generalize several previous results. We also extend our to infinite discretizations unbounded consider algorithmic aspects. can be applied component topology preservation as well extracting geometric features connectivity control very large object discretizations.