Spacetimes as topological spaces, and the need to take methods of general topology more seriously

TL;DR

This paper critically reviews the use of topological methods in spacetime physics, questioning the choice of topology and advocating for more rigorous application of general topology to better understand spacetime singularities and causal structures.

Contribution

It provides a comprehensive survey and critique of current topological approaches in spacetime theory, highlighting the need for integrating methods from general topology.

Findings

Questions the adequacy of manifold topology for spacetime description

Highlights the importance of causal structure in topology choice

Identifies open problems in topological modeling of spacetime

Abstract

Why is the manifold topology in a spacetime taken for granted? Why do we prefer to use Riemann open balls as basic-open sets, while there also exists a Lorentz metric? Which topology is a best candidate for a spacetime; a topology sufficient for the description of spacetime singularities or a topology which incorporates the causal structure? Or both? Is it more preferable to have a topology with as many physical properties as possible, whose description might be complicated and counterintuitive, or a topology which can be described via a countable basis but misses some important information? These are just a few from the questions that we ask in this Chapter, which serves as a critical review of the terrain and contains a survey with remarks, corrections and open questions.