Integrating Space, Time, Version, and Scale using Alexandrov Topologies

As a contribution to higher dimensional spatial data modelling this article introduces a novel approach to spatial database design. Instead of extending the canonical Solid-Face-Edge-Vertex schema of topological data, these classes are replaced altogether by a common type SpatialEntity, and the individual “bounded-by” relations between two consecutive classes are replaced by one separate binary relation BoundedBy on SpatialEntity. That relation defines a so-called Alexandrov topology on SpatialEntity and thus exposes the fundamental mathematical principles of spatial data design. This has important consequences: First, a formal definition of topological “dimension” for spatial data can be given. Second, every topology for data of arbitrary dimension has such a simple representation. Also, version histories have a canonical Alexandrov topology, and generalisations can be consistently modelled by the new consistency rule continuous functions between LoDs, and monotonicity enables accelerated path queries. The result is a relational database schema for spatial data of dimension 6 and more which seamlessly integrates 4D space-time, levels of details and version history. Topological constructions enable queries across these different aspects.