On the Structure of Topological 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 spatial. A special class of spatial fibrous preorders consisting of an interconnected family of preorders indexed by a unitary magma is called Cartesian and is studied here. Topological spaces that are obtained from those fibrous preorders, with a unitary magma I, are called I-Cartesian and are characterized. The characterization reveals a hidden structure on 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.