TITLE: Cardinality of connected manifolds
QUESTION [11 upvotes]: Consider the assertion:
Every connected, but not necessarily paracompact, n-manifold is of cardinality 
$2^{\aleph_0}$ (at least assuming the axiom of choice). 
For n=1 this may be proved via enumeration of the short list of examples. The essential point is that while there is a Long Line, there is no Extra Long Line.
What is the situation for n>1?

REPLY [5 votes]: Stephen's argument can also be phrased in the language of model theory. Given a connected manifold $X$, consider an elementary submodel of a large fragment of set theory $\mathfrak{M}$ that (1) contains all the reals,
(2) is closed under countable subsets, (3) contains $X$ as an element
and (4) is of size $2^{\aleph_0}$.
It suffices to show that $X\subseteq \mathfrak{M}$. But $X\cap \mathfrak{M}$ is open since each point of $X$ has a neighbourhood of size $2^{\aleph_0}$ and, by elementarity there is a bijection from the reals to this neighbourhood and, since the $\mathfrak M$ contains the reals it must also contain the image of this bijection, namely the neighbourhood. But $X\cap \mathfrak{M}$ is also closed since $\mathfrak M$ contains all sequences from $X\cap \mathfrak{M}$ and hence their unique (by Hausdorffness) limits. By connectedness $X\cap \mathfrak{M} = X$.