Title: Topology and experimental distinguishability,
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.