Experimental verifiability and topology

TL;DR

The paper explores how topological spaces and sigma-algebras underpin experimental verifiability in physics, linking mathematical structures to physical distinguishability and constraining scientific theories.

Contribution

It demonstrates that experimental distinguishability requires a Kolmogorov and second countable topology, providing a foundational link between topology and scientific verifiability.

Findings

Topological spaces are essential for experimental distinguishability.

Experimental objects form a $T_0$ and second countable topology.

Mathematical structures gain physical meaning through these topological constraints.

Abstract

We briefly show how the use of topological spaces and $\sigma$-algebras in physics can be rederived and understood as the fundamental requirement of experimental verifiability. We will see that a set of experimentally distinguishable objects will necessarily be endowed with a topology that is Kolmogorov (i.e. $T_0$) and second countable, which both puts constraints on well-formed scientific theories and allows us to give concrete physical meaning to the mathematical constructs. These insights can be taken as a first step in a general mathematical theory for experimental science.