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).
- 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.
- 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?