To characterize large separable spaces in which every non-trivial connected subset has non-empty interior?

$\mathbb R$ with usual metric is a space such that every connected subset with more than one point has non-empty interior . My question is , can we characterize those separable metric spaces with more than one point in which every connected subset with more than one point has non-empty interior ? Or more strongly , can we characterize those separable , infinite metric spaces in which every infinite connected subset has non-empty interior ? More restrictively can we characterize those separable , uncountable metric spaces in which every uncountable connected subset has non-empty interior ? And what if we loosen all metric spaces with topological spaces ?

• Note that $\Bbb R^2$ is not such a space, thanks to the Knaster-Kuratowski fan. – Brian M. Scott Oct 3 '15 at 15:13