Path-connectedness in the cofinite topology

I recently proved the following for a homework exercise.

Let $$X$$ be an infinite set endowed with the cofinite topology (open, iff finite complement or empty).

1. If the cardinality of a subset $$U\subseteq X$$ is at least that of $$\mathbb R$$ (i.e. there exists an injection $$\mathbb R\to U$$), then $$U$$ is path-connected.
2. If an infinite subset $$U\subseteq X$$ is path-connected, then it is uncountable.

Those two statements kind of look like an equivalence to me, if only I could reduce that "cardinality $$\geq\mathbb R$$" to "uncountable". My question now is, whether every uncountable subset of $$X$$ necessarily is path-connected or not? If the continuum hypothesis is assumed to be true (i.e. there does not exist any set with cardinality strictly between $$\mathbb N$$ and $$\mathbb R$$), then this follows trivially from 1., but otherwise I have no clue. Does it maybe depend on the continuum hypothesis?