  We present a 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. Our results are independent
of whether the empty class is a set and whether atoms exist.
  This article is a revised version of the first part of the author's doctoral
thesis.
