  We relate closure operations for ideals and for submodules to non-flat
Grothendieck topologies. We show how a Grothendieck topology on an affine
scheme induces a closure operation in a natural way, and how to construct for a
given closure operation fulfilling certain properties a Grothendieck topology
which induces this operation. In this way we relate the radical to the
surjective topology and the constructible topology, the integral closure to the
submersive topology, to the proper topology and to Voevodsky's h-topology, the
Frobenius closure to the Frobenius topology and the plus closure to the finite
topology. The topologies which are induced by a Zariski filter yield the
closure operations which are studied under the name of hereditary torsion
theories. The Grothendieck topologies enrich the corresponding closure
operation by providing cohomology theories, rings of global sections, concepts
of exactness and of stalks.
