Synthetic Topology and Constructive Metric Spaces

TL;DR

This thesis explores synthetic topology within metric spaces, redefining the framework to include closed sets, reconstructing real numbers, and analyzing models where metric and intrinsic topologies align.

Contribution

It introduces a revised synthetic topology model incorporating closed sets and reconstructs real numbers as open Dedekind cuts, expanding the theoretical foundation.

Findings

Redefinition of synthetic topology to include closed sets

Reconstruction of real numbers as open Dedekind cuts

Analysis of models where metric and intrinsic topologies match

Abstract

The thesis presents the subject of synthetic topology, especially with relation to metric spaces. A model of synthetic topology is a categorical model in which objects possess an intrinsic topology in a suitable sense, and all morphisms are continuous with regard to it. We redefine synthetic topology in order to incorporate closed sets, and several generalizations are made. Real numbers are reconstructed (to suit the new background) as open Dedekind cuts. An extensive theory is developed when metric and intrinsic topology match. In the end the results are examined in four specific models.