Can a connected second countable manifold have cardinality $>2^{\aleph_0}$?

Question

Can a connected second countable manifold have cardinality $>2^{\aleph_0}$?

From here, a connected Hausdorff manifold must have cardinality $2^{\aleph_0}$. How about if we change the condition "Hausdorff" to "second countable"?

Answer 1

Regardless of connectedness: since the manifold $M$ is second-countable, it has a countable atlas (every open covering of a second-countable space admits a countable subcovering). Therefore, it is countable union of subsets homeomorphic to the unit ball of $\Bbb R^n$, whose cardinality is at most $2^{\aleph_0}$.

Therefore its cardinality is $\le \aleph_0\cdot 2^{\aleph_0}=2^{\aleph_0}$.