Apparently, there is the following fact:



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> The set of homeomorphism classes of connected manifolds has the same cardinality $c$ as that of $\mathbb R$.
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I find it to be interesting; but would be happier to see a proof, and would be grateful for a reference somewhere.



Counting connected manifolds
Upper bound (assuming the manifolds are second countable): every manifold admits a complete metric, and the "set" of isometry classes of complete separable metric spaces is of cardinality continuum. Indeed, every such a space is a completion of a countable metric space, and there is only continuum of metrics on a countable set.


Lower bound: there is a continuum of non-isomorphic fundamental groups of manifolds. E.g. any countable sum of cyclic groups is a first homology group of a connected manifold: just take a connected sum of an appropriate countable collection of lens spaces.


