Analytic cohomology of a complex manifold modelled on Banach spaces can be topologized, and the right choice is the countable-cover (tau-delta) topology, the unique one among the several reasonable candidates that is a direct limit of Banach space topologies and admits a topological Leray theorem. For a locally Stein manifold, a separated cohesive sheaf, and a cover by Stein open sets, the canonical map from the Cech cohomology of the cover to the cohomology of the manifold is an isomorphism of topological vector spaces - continuity and bijectivity being classical, with the openness of the Cech differential the substantive step. Separatedness is necessary: a cohesive but nonseparated sheaf over the plane already breaks the conclusion in degree zero.
Laszlo Lempert (arXiv 1312.7327v1, math.CV, December 2013; MSC 32C35, 46E50, 58B12; supported by NSF grants DMS0700281 and 1162070) puts a topology on the analytic cohomology groups of infinite-dimen…