A Completeness Theorem for Topological Doctrines

TL;DR

This paper develops a completeness theorem for topological doctrines by extending logical categories with specific operators and axioms, enabling an embedding into powers of topological spaces.

Contribution

It introduces new axioms and operators to logical categories, establishing an embedding theorem into topological space powers.

Findings

Extended logical categories with fiberwise interior and closure operators.

Proved an embedding theorem into powers of topological spaces.

Identified key axioms like product independence and loop contraction.

Abstract

We extend logical categories with fiberwise interior and closure operators so as to obtain an embedding theorem into powers of the category of topological spaces. The required axioms, besides the Kuratowski closure axioms, are a `product independence' and a `loop contraction' principle.