Jens Mammen, a psychologist, has proposed a model of the human mind based on the idea that the brain organizes objects in the world into two kinds of general categories: Broad categories, which he called "sense categories", and categories of special, distinguished objects (or people), which he called "choice categories".

From a mathematical point of view, it is interesting that Mammen formulated his model of the mind axiomatically, based on the notion of a topological space. The objects in the universe are modelled by the points in a topological space (U,S), where the (broad) sense categories are modelled by open sets in the topology S. The choice categories forms an additional collection of subsets of the universe, C, that together with the topology must adhere to certain axioms. The triple (U,S,C) is called a "Mammen space" (a term that I introduced).

Several mathematical questions arise out Mammen's theory. For instance, if we want Mammen's model to be able to account for all possible subsets of the universe (a property Mammen called "completeness"), then the Axiom of Choice, or at least some non-trivial consequences thereof, seem to play a role. There are also several interesting questions related to cardinal invariants, such as the "weight" of the underlying topological space of a complete Mammen space.

I will give an overview of the mathematics of Mammen spaces and known results, and also discuss the numerous unsolved problems that remain.