Study of Topological Spaces that Don’t Rely on Points as their Basis

Abstract

Traditionally, the concept of points and neighborhood structures has been quite dominant in defining and further analyzing various properties in the study of topological spaces. In pointless topology-or point-free topology, also referred to as locale theory-the key emphasis is shifted from points to lattices of open sets. In addition to gaining further insight into spatial properties, this approach has important implications in fields like theoretical computer science, logic, and categorical theory. It is the purpose of the present work to be mainly interested in issues of the foundation of point-free topology: locales, frames, mutual relationships between them; the ways of interpreting classical topological concepts in the new language and the strong and weak points of such a point-free view. In the process, we focus on some of the significant applications of point-free topology, both within mathematics and beyond. The present paper is an attempt to give a broad overview of pointfree topology, discussing both its theoretical basis and its practical significance.

Keywords

• point-free topology, locale theory, frames, lattices of opens, spatiality, topological spaces, the categorical approach, theoretical computer science, logics, pointless topology