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

Experimental verifiability and topology