Frames and topological spaces in MSet

In this paper we give a definition for frames in the category MSet of actions of a monoid M on sets. We then show that, like in the classical case, frames and (internally) complete Heyting algebras are the same. Further, we give a definition of a topological space in MSet and study the relation between frames and topological spaces in this category. We show that the well known adjunction between the two categories still exists in MSet.