The Topology of Shapes Made with Points

TL;DR

This paper explores the topological properties of point-based shapes in design, revealing their equivalence to finite spaces and connecting their topology to preorder relations, with implications for understanding point-free shapes.

Contribution

It establishes that shapes made with points are mathematically equivalent to finite spaces, linking their topology to preorder relations and comparing them with point-free shapes.

Findings

Shapes made with points are equivalent to finite topological spaces.

Topology of point-based shapes corresponds to preorder relations.

Connections between point-based and point-free shape topologies are discussed.

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 $U_i$, 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" pictorial shapes (when $i > 0$) are discussed and the main differences between the two are summarized.