Inexistence of Zeeman's fine topology

TL;DR

This paper proves that Zeeman's fine topology and its extensions do not exist, analyzing the reasons and implications of their non-existence within the context of spacetime topologies.

Contribution

It demonstrates the non-existence of Zeeman's fine topology and its extensions, clarifying the limitations of certain spacetime topologies in mathematical physics.

Findings

Zeeman's fine topology does not exist.

G"obel's extension to arbitrary spacetimes does not exist.

The family of topologies has no maximal element.

Abstract

The family of topologies that induce the Euclidean metric space on every time axis and every space axis exhibits no maximal element when partially ordered by the relation ``finer than'', as demonstrated in this article. One conclusion and two reflections emerge and are addressed herein: Conclusion: a. Zeeman's fine topology [1] and G\"{o}bel's extension to arbitrary spacetimes [2] do not exist. Reflections: a. Both authors' attempts may be classified as type-2 strategies, within the taxonomy of [3]. b. How could these inexistent topologies be used for decades?