  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.
