Topological set theories and hyperuniverses

We give a new set theoretic system of axioms motivated by a
topological intuition: The set of subsets of any set is a topology
on that set. On the one hand, this system is a common weakening of
Zermelo-Fraenkel set theory ZF, the positive set theory GPK and the
theory of hyperuniverses. On the other hand, it retains most of the
expressiveness of these theories and has the same consistency
strength as ZF. We single out the additional axiom of the universal
set as the one that increases the consistency strength to that of
GPK and explore several other axioms and interrelations between
those theories. Hyperuniverses are a natural class of models for
theories with a universal set. The Aleph_0- and Aleph_1-dimensional
Cantor cubes are examples of hyperuniverses with additivity
Aleph_0, because they are homeomorphic to their hyperspace. We
prove that in the realm of spaces with uncountable additivity, none
of the generalized Cantor cubes has that property. Finally, we give
two complementary constructions of hyperuniverses which generalize
many of the constructions found in the literature and produce
initial and terminal hyperuniverses.