  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. This work is an
overview of some of the results of Assumptions of Physics, a project that aims
to identify a handful of physical principles from which the basic laws can be
rigorously derived (see https://assumptionsofphysics.org ).
