Topology and experimental distinguishability

TL;DR

This paper proposes that topology's main role in experimental sciences is to formalize what can be distinguished through experiments, linking topological concepts to experimental distinguishability and observation.

Contribution

It introduces a framework connecting experimental observations with topological structures, justifying the use of topology in science based on distinguishability.

Findings

Observations form a topology on the set of elements.

Experimental relationships are continuous functions.

The collection of experimental relationships is itself topologically distinguishable.

Abstract

In this work we introduce the idea that the primary application of topology in experimental sciences is to keep track of what can be distinguished through experimentation. This link provides understanding and justification as to why topological spaces and continuous functions are pervasive tools in science. We first define an experimental observation as a statement that can be verified using an experimental procedure and show that observations are closed under finite conjunction and countable disjunction. We then consider observations that identify elements within a set and show how they induce a Hausdorff and second-countable topology on that set, thus identifying an open set as one that can be associated with an experimental observation. We then show that experimental relationships are continuous functions, as they must preserve experimental distinguishability, and that they are themselves experimentally distinguishable by defining a Hausdorff and second-countable topology for this collection.