Can a connected finite-dimensional manifold have cardinality $>2^{\aleph_0}$?

Question

Can a connected finite-dimensional manifold have cardinality $>2^{\aleph_0}$?

I know that if we either impose the condition "Hausdorff" or "second countable", the assertion is false. What if we drop these two common requirements?

Answer 1

There are connected (non-Hausdorff, non-second countable) manifolds of arbitrarily large cardinality. For instance, let $S$ be any discrete space and let $X$ be the quotient of $\mathbb{R}\times S$ by the equivalence relation that identifies $(a,s)$ with $(a,t)$ whenever $a\neq 0$. (This is like the well-known "line with two origins", except with $|S|$ origins instead of just two.) Then $X$ is connected and locally homeomorphic to $\mathbb{R}$, but the points $[0,s]$ are distinct for all $s\in S$, so $|X|\geq|S|$. In fact, $|X|=|S|+2^{\aleph_0}$, so you can get $X$ to have any cardinality $\geq 2^{\aleph_0}$.

(We can also make $X$ a smooth or even real-analytic manifold by taking the maps $\mathbb{R}\cong\mathbb{R}\times\{s\}\to\mathbb{R}\times S\to X$ as charts for each $s\in S$.)