Generalisation of Berge's Maximum Theorem

A categorical approach to the maximum theorem

In this paper we state and prove the maximum theorem in terms of the theory of monoidal topology and the theory of double categories.This approach allows us to generalise (the main assertion of) the maximum theorem, which is classically stated for topological spaces, to pseudootopological spaces and pretopological spaces, as well as to closurespaces, approach spaces and probabilistic approach spaces, amongst others.As a part of this we prove a generalisation of the extreme value theorem.