  It was realized early on that topologies can model constructive systems, as
the open sets form a Heyting algebra. After the development of forcing, in the
form of Boolean-valued models, it became clear that, just as over ZF any
Boolean-valued model also satisfies ZF, so do Heyting-valued models satisfy
IZF, which stands for Intuitionistic ZF, the most direct constructive
re-working of the ZF axioms. In this paper, we return to topologies, and
introduce a variant model, along with a correspondingly revised forcing or
satisfaction relation. The purpose is to prove independence results related to
weakenings of the Power Set axiom. The original motivation is earlier work,
based on the set of reals, which shows that Exponentiation, in the context of
CZF minus Subset Collection, does not suffice to prove that the Dedekind reals
form a set. The current semantics is the generalization of that model from R to
an arbitrary topological space. It is investigated which set-theoretic
principles hold in such models in general. In addition, natural properties of
the underlying topological space are shown to imply the validity of stronger
such principles.
