Does Zeeman's Fine Topology Exist?

TL;DR

This paper investigates the existence of Zeeman's fine topology on Minkowski space, analyzing the lattice of topologies that induce Euclidean metrics on axes, and concludes that its supremum does not belong to this family.

Contribution

It provides a lattice-theoretic analysis of topologies related to Zeeman's fine topology and demonstrates that its supremum is not contained within the considered family.

Findings

The supremum of the sublattice Z does not belong to Z.

Zeeman's fine topology is not the maximum in the lattice of topologies inducing Euclidean metrics.

Mathematical and physical implications of the non-existence of the supremum are discussed.

Abstract

We work on the family of topologies for the Minkowski manifold M. We partially order this family by inclusion to form the lattice \Sigma(M), and focus on the sublattice Z of topologies that induce the Euclidean metric space on every time axis and every space axis. We analyze the bounds of Z in the lattice \Sigma(M), in search for its supremum. Our conclusion --that such a supremum does not belong in Z-- is compared with constructive proofs of existence of the fine topology, defined as the maximum of Z and conceived to play an essential role in contemporary physical theories. Essential mathematical and physical questions arise.