On the structure of topological spaces

TL;DR

This paper explores the structure of topological spaces through fibrous preorders, introducing concepts like spacial and cartesian fibrous preorders, and characterizes spaces derived from these structures, including metric and normed spaces.

Contribution

It introduces the concepts of spacial and cartesian fibrous preorders, providing new characterizations of topological spaces and linking them to algebraic structures like monoids and groups.

Findings

Characterization of I-cartesian spaces reveals hidden structures.

Procedures to derive topologies from algebraic data types.

Application to metric and normed spaces.

Abstract

The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spacial. A special class of spacial fibrous preorders consisting of an interconnected family of preorders indexed by a unitary magma is called cartesian and studied here. Topological spaces that are obtained from those fibrous preorders, with a unitary magma \emph{I}, are called \emph{I}-cartesian and characterized. The characterization reveals a hidden structure of such spaces. Several other characterizations are obtained and special attention is drawn to the case of a monoid equipped with a topology. A wide range of examples is provided, as well as general procedures to obtain topologies from other data types such as groups and their actions. Metric spaces and normed spaces are considered as well.