Topologimeter and the problem of physical interpretation of topology lattice

TL;DR

This paper introduces the topologimeter, a quantum-like observable for topologies on three points, exploring its physical interpretation and drawing analogies with Stern-Gerlach experiments.

Contribution

It proposes a novel quantum-like framework for analyzing topologies, highlighting asymmetries and experimental analogies.

Findings

Topologimeter can be treated as a quantum-like observable.

Asymmetry between bra- and ket-spaces is essential.

Analogies with Stern-Gerlach experiments are established.

Abstract

The collection of all topologies on the set of three points is studied treating the topology as quantum-like observable. It turns out to be possible under the assumption of the asymmetry between the spaces of bra- and ket-vectors. The analogies between the introduced topologimeter and the Stern-Gerlach experiments are outlined.