  Berge's maximum theorem gives conditions ensuring the continuity of an
optimised function as a parameter changes. 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
pseudotopological spaces and pretopological spaces, as well as to closure
spaces, approach spaces and probabilistic approach spaces, amongst others. As a
part of this we prove a generalisation of the extreme value theorem.
