Are there countably many closed manifolds in each dimension?

Question

There is a single closed topological 1-manifold (up to, of course, homeomorphism): $S^1$. The classification of surfaces shows that there are countably many closed topological 2-manifolds. Classification of closed, orientable topological 3-manifolds reveals that there are also a countable number of these, though I am unclear in the non-orientable case. Does every dimension admit only a countable number of closed topological manifolds?

Answer 1

Andreas Thom answered this question on MathOverflow here:

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.