Preservation Theorems Through the Lens of Topology

TL;DR

This paper introduces a new topological framework to analyze preservation theorems, enabling a unified study of their relativization and structural properties within topology.

Contribution

It develops a novel family of topological spaces that encapsulate preservation theorems and integrates existing results into this unified framework.

Findings

New topological spaces capturing preservation theorems

Framework for relativization under various morphisms and constructions

Unification of known results within the new topological setting

Abstract

In this paper, we introduce a family of topological spaces that captures the existence of preservation theorems. The structure of those spaces allows us to study the relativisation of preservation theorems under suitable definitions of surjective morphisms, subclasses, sums, products, topological closures, and projective limits. Throughout the paper, we also integrate already known results into this new framework and show how it captures th essence of their proofs.