The Order on the Light Cone and its induced Topology

TL;DR

This paper clarifies misconceptions about a topology related to the light cone in spacetime, showing its relation to Zeeman topologies and emphasizing the need for further study of spacetime topology.

Contribution

It corrects a recent misconception, characterizes the topology as an intersection topology involving the light cone, and links it to Zeeman topologies.

Findings

The topology is an intersection of Euclidean and order topologies.

The topology is a Zeeman topology.

Highlights the importance of studying spacetime topologies systematically.

Abstract

In this article we first correct a recent misconception about a topology that was suggested by Zeeman as a possible alternative to his Fine topology. This misconception appeared while trying to establish the causality in the ambient boundary-ambient space cosmological model. We then show that this topology is actually the intersection topology (in the sense of G.M. Reed) between the Euclidean topology on $\mathbb{R}^4$ and the order topology whose order, namely horismos, is defined on the light cone. Last, but not least, we show that the order topology from horismos belongs to the class of Zeeman topologies. These results accelerate the need for a deeper and more systematic study of the global topological properties of spacetime manifolds.