Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be triangularized and hence be described by a finite amount of combinatorial data.

In higher dimensions this argument doesn't work anymore. So it still true for $n\ge 4$ that the set of homeomorphism classes of compact, connected topological $n$-manifolds (without boundary) is countable?

(I'd be also interested in the same question for diffeomorphism classes of compact smooth manifolds.)

diffeomorphismclasses the answer is in the affirmative, as any smooth manifold can be triangulated and within each $PL$-class there are finitely many diffeomorphism classes. $\endgroup$4more comments