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.)
Best Answer
It was shown in
J. Cheeger and J. M. Kister, Counting topological manifolds. Topology 9, 1970 149–151.
that there are only countably many compact manifolds up to homeomorphism (even allowing boundaries).
Here is a link to the article.