State property systems and closure spaces: A study of categorical equivalence

We show that the natural mathematical structure to describe a physical entity by means of its states and its properties within the Geneva-Brussels approach is that of a state property system. We prove that the category of state property systems (and morphisms) SP is equivalent to the category of closure spaces (and continuous maps) Cls. We show the equivalence of the 'state determination axiom' for state property systems with the 'T-0 separation axiom' for closure spaces. We also prove that the category SP0 of state-determined state property systems is equivalent to the category L-0 of based complete lattices. In this sense the equivalence of SP and Cls generalizes the equivalence of Cls(0) (T-0 closure spaces) and L-0 proven by Erne (1984).