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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?

Plutoro
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  • The comments to this question http://math.stackexchange.com/questions/109881/can-we-prove-that-there-are-countably-many-isomorphism-classes-of-compact-lie-gr imply that there are only countably many closed manifolds in each dimension. Perhaps you could distill them into one coherent response? – Jason DeVito - on hiatus Apr 04 '16 at 17:16
  • http://mathoverflow.net/q/198098/40804 –  Apr 04 '16 at 17:18
  • @MikeMiller Perfect. Thanks. – Plutoro Apr 04 '16 at 17:20
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    The answer is yes, proved by Cheeger and Kister - see http://mathoverflow.net/questions/198098/are-there-only-countably-many-compact-topological-manifolds. – Noah Schweber Apr 04 '16 at 17:21

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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.

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