Point-set topological spatial relations

TLDR

GIS practical needs have driven the search for formal, sound methods to describe spatial relations. The paper develops a novel theory of topological spatial relations between sets based on boundary and interior intersections. By classifying intersection emptiness, the authors define sixteen relations, which collapse to nine for spatial regions in a connected topological space, all realizable in ℝ². These relations map onto standard set‑theoretic and topological concepts such as equality, disjointness, and interior containment.

Abstract

Abstract Practical needs in geographic information systems (GIS) have led to the investigation of formal and sound methods of describing spatial relations. After an introduction to the basic ideas and notions of topology, a novel theory of topological spatial relations between sets is developed in which the relations are defined in terms of the intersections of the boundaries and interiors of two sets. By considering empty and non-empty as the values of the intersections, a total of sixteen topological spatial relations is described, each of which can be realized in R 2. This set is reduced to nine relations if the sets are restricted to spatial regions, a fairly broad class of subsets of a connected topological space with an application to GIS. It is shown that these relations correspond to some of the standard set theoretical and topological spatial relations between sets such as equality, disjointness and containment in the interior.