TITLE: Is a cofinite topology for a set with cardinality between $\aleph_{0}$ and $2^{\aleph_{0}}$ path-connected?
QUESTION [8 upvotes]: It is easy to show that $\mathbb{N}$ with the cofinite topology is not path connected and that any set with cardinality $\geq 2^{\aleph_0}$ equipped with the cofinite topology is  in fact  path connected.
what about cardinalities $\aleph_0<\alpha<2^{\aleph_0}$ (under the assumption that such exist obviously)?
If $\alpha $ is path connected then any cardinality $\geq \alpha$ is also, so an interesting direction would be trying and checking whether $\aleph_1$ is path connected (I have no clue about how one can even start checking this).

REPLY [5 votes]: A continuous non-constant function from $[0,1]$ into $X$ with the cofinite topology exists iff $[0,1]$ has a partition into $\le |X|$ many disjoint closed non-empty subsets. 
This question discusses the options for the cardinality of such a partition. One of the conclusions is that under $\textrm{MA}(\omega_1)$ we have that a set of size $\aleph_1$ in the cofinite topology is (connected and ) not path-connected, while of course under CH such a set is path-connected. So the case $\aleph_1$ is undecidable under ZFC.