Journal Article
The Topology of Shapes made with Points

In architecture, city planning, visual arts, and other design areas, shapes are often made with points, or with structural representations based on point-sets. Shapes made with points can be understood more generally as finite arrangements formed with elements (i.e. points) of the algebra of shapes Ui, for i = 0. This paper examines the kind of topology that is applicable to such shapes. From a mathematical standpoint, any "shape made with points" is equivalent to a finite space, so that topology on a shape made with points is no different than topology on a finite space: the study of topological structure naturally coincides with the study of preorder relations on the points of the shape. After establishing this fact, some connections between the topology of shapes made with points and the topology of point-free shapes (when i > 0) are discussed and the main differences between the two are summarized.

Title
Publication TypeJournal Article
Year of Publication2019
AuthorsCharidis A
JournalEnvironment and Planning B: Urban Analytics and City Science
VolumeOnline First
Date Published02/2019
Type of ArticleResearch Article
KeywordsFinite Order Topology, Shape with Points, Structural Description, T0-Space
Abstract

In architecture, city planning, visual arts, and other design areas, shapes are often made with points, or with structural representations based on point-sets. Shapes made with points can be understood more generally as finite arrangements formed with elements (i.e. points) of the algebra of shapes Ui, for i = 0. This paper examines the kind of topology that is applicable to such shapes. From a mathematical standpoint, any "shape made with points" is equivalent to a finite space, so that topology on a shape made with points is no different than topology on a finite space: the study of topological structure naturally coincides with the study of preorder relations on the points of the shape. After establishing this fact, some connections between the topology of shapes made with points and the topology of point-free shapes (when i > 0) are discussed and the main differences between the two are summarized.

URLhttps://doi.org/10.1177/2399808319827015
DOI10.1177/2399808319827015