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). 



Is a cofinite topology for a set with cardinality between $\aleph_{0}$ and $2^{\aleph_{0}}$ path-connected?
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](https://mathoverflow.net/q/285780/2060) 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.


