Is a cofinite topology for a set with cardinality between $\aleph_{0}$ and $2^{\aleph_{0}}$ path-connected?

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).

• It's classical that every nonempty Hausdorff compact perfect space has cardinal $\ge c$. The image of a non-constant path in a Hausdorff compact space satisfies these assumptions, and hence has cardinal $\ge c$. Hence, every Hausdorff compact space of cardinal $<c$ is totally path-disconnected.
– YCor
Jul 5 '19 at 19:21
• (No need of compactness in the last sentence of my last comment: the consequence is: every Hausdorff space of cardinal $<c$ is totally path-disconnected.")
– YCor
Jul 5 '19 at 19:50
• @YCor: The space in question isn't Hausdorff, though. In fact, I think that the cofinite topology on some cardinal $\alpha$ should be path connected provided that $\alpha$ is $\geq$ the cardinal $\acute{\mathfrak{n}}$ from this question: mathoverflow.net/questions/285780/…. The idea is that if $\{X_\xi \,:\, \xi < \alpha \}$ is a partition of $[0,1]$ into closed sets, then the mapping that sends $X_\xi$ to $\xi$ is continuous (when $\alpha$ has the cofinite topology). Jul 5 '19 at 20:08
• @PietroMajer I only know that your first statement is true for Hausdorff spaces. The line with origin doubled is T1 and path-connected, but not arc-connected. Jul 5 '19 at 20:40
• @WillBrian In fact, this is if and only if, isn't it? Jul 6 '19 at 5:45

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.