Let $X$ be a compact, connected Hausdorff space with at least two points.

In $\mathrm{ZF}+\mathrm{AC}_\omega(\mathbb R)$, $X$ is metrizable, and from this it can be shown that $X$ is uncountable.

In $\mathrm{ZF}$, however, the space may not be metrizable. Does anyone know if it's still possible to prove that it is uncountable?