Let $M$ be a connected closed smooth manifold. Are there at most countably many non-diffeomorphic symplectic forms in any given class in $H^2(M, \mathbb{R})$?



At most countably many symplectic forms in given cohomology class
That's true and follows from the fact a vector space with a countable dense subset can't have an uncountable number of open subsets that don't intersect pairwise. Here the vector space is the space of all $C^{\infty}$ $2$-forms in the given cohomology class and the open subsets are equivalence classes of non-diffeomorphic symplectic forms. It's easy to construct a countable subset of $2$-forms, dense in $ C^{\infty}$ topology.


