Local and global topology preservation in locally finite sets of tiles

This paper deals with sets V of tiles (compact, convex sets) in R. Tiles are a generalization of pixels or voxels (in R or R); they can have arbitrary shapes and are allowed to overlap. The union of all the tiles of V is denoted by U(V). The neighborhood N-p(P) of a tile P is the union of the tiles of V that intersect V. P is called simple if deletion of P from V does not change the topology (in the homotopy sense) of U(V). We show in this paper that if V satisfies a property called strong normality (SN), and deletion of P preserves the topology of N-p(P), then P is simple. This may not be true if V is not SN; and even if V is SN, P may be simple even if deletion of P does not preserve the topology of N-p(P).