Problem of determining whether a parallel reduction operator for n -dimensional binary images always preserves topology

Loosely speaking, a simple set of a finite binary image is a set of 1s whose deletion `preserves topology.' This concept can be made precise in different (and inequivalent) ways. Ronse established results which imply that, for finite 2-D binary images on a Cartesian grid and three different definitions of simple set, a set S of 1s is simple if every subset of S that lies in a 2- point by 2-point square is simple. In fact this is a special case of a general result which applies to arbitrary finite binary images -- not just 2-D images on a Cartesian grid -- and any definition of simple set which satisfies three axioms stated in this paper. For finite binary images on an n-dimensional Cartesian grid, we give appropriate definitions of simple set which satisfy all the axioms. When these definitions of simple set are used, verification that a parallel reduction operator for n-dimensional binary images preserves the topology of all possible input images may be achievable by checking only a finite number of cases.