Q:

Manifold that's not $\aleph_1$-separable

I know that there are manifolds that are not separable, such as the long line $\omega_1 \times [0, 1) \setminus \{(0, 0)\}$. Is there any manifold with no uncountable dense set of cardinality $\aleph_1$?
A line $\omega_2 \times [0, 1) \setminus \{(0, 0)\}$ wouldn't work since no neighborhoods of $(\omega_1, 0)$ are homeomorphic to $\mathbb{R}$. Would something else work?

A:

It is at least consistent that every manifold have a dense subset of cardinality $\omega_1$: every manifold has cardinality $2^\omega=\mathfrak{c}$, so under $\text{CH}$ every manifold has cardinality $\omega_1$. (This is Theorem 2.9 in Peter Nyikos, The Theory of Nonmetrizable Manifolds, in Handbook of Set-Theoretic Topology, K. Kunen & J.E. Vaughan, eds., North Holland, 1984.) However, in section 3.7 he constructs a manifold of cellularity (and therefore density) $\mathfrak{c}$; under $\neg\text{CH}$ it has no dense subset of cardinality $\omega_1$.

