  In the present paper, we give a systematic study of the correspondence theory
of generalized modal algebras and generalized modal spaces. The special feature
of the present paper is that in the proof of the (right-handed) topological
Ackermann lemma, the admissible valuations are not the clopen valuations
anymore, but values in the set DK(X) which are only closed and satisfy
additional properties, not necessarily open. This situation is significantly
different from existing settings using Stone/Priestley-like dualities, where
all admissible valuations are clopen valuations.
