  For open and singular varieties in positive characteristic p we study the
existence of an integral p-adic cohomology theory which is finitely generated,
compatible with log crystalline cohomology and rationally compatible with rigid
cohomology. We develop such a theory under certain assumptions of resolution of
singularities in positive characteristic, by using cdp- and cdh-topologies.
  Without resolution of singularities in positive characteristic, we prove the
existence of a good p-adic cohomology theory for open and singular varieties in
cohomological degree 1, by using split proper generically \'etale
hypercoverings. This is a slight generalisation of a result due to
Andreatta--Barbieri-Viale. We also prove that this approach does not work for
higher cohomological degrees.
