We can use the topological properties of interiors and boundaries to define relationships between spatial features. Since the properties of interiors and boundaries do not change under topological mapping, we can investigate their possible relations between spatial features. We can define the interior of a region, R, as the largest set of points of R for which we can construct a disc-like environment around it (no matter how small) that also falls completely inside R. The boundary of R is the set of those points belonging to R that do not belong to the interior of R, i.e. one cannot construct a disc-like environment around such points that still belongs to R completely.
Let us consider a spatial region A. It has a boundary and an interior, both seen as (infinite) sets of points, which are denoted by boundary(A) and interior(A), respectively. We consider all possible combinations of intersections (∩) between the boundary and the interior of A with those of another region, B, and test whether they are the empty set (∅) or not. From these intersection patterns, we can derive eight (mutually exclusive) spatial relationships between two regions. If, for instance, the interiors of A and B do not intersect, but their boundaries do, yet the boundary of one does not intersect the interior of the other, we say that A and B meet.