How many manifolds are there?

Question

I think there are three questions here, an answer to any would be interesting:

1. How many compact manifolds are there?
2. How many (not necessarily compact) manifolds are there?
3. How many compact/not-compact manifolds of dimension $n$ are there?

I know there exist uncountably many smooth structures on $\Bbb R^4$, so there are up to diffeomorphism uncountably many smooth manifolds. I also know there are only countably many compact $2$d manifolds.

Here I would view to manifolds as being the same if there is a homeomorphism between them, but I would restrict to smooth manifolds and not to topological manifolds. These definitions are basically arbitrary, so I don't really mind if you take a different view.

Answer 1

1. There are countably many compact topological manifolds (up to homeomorphism). This is actually quite nontrivial, proven in

J. Cheeger and J. M. Kister, Counting topological manifolds, Topology, vol. 9, (1970) p. 149–151.

2. Continuum of topological manifolds (up to homeomorphism). This is so already for surfaces.

3. Same as (1) and (2).