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$\mathbb R$ 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 ?

Please help . Thanks in advance

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  • $\begingroup$ Note that $\Bbb R^2$ is not such a space, thanks to the Knaster-Kuratowski fan. $\endgroup$ – Brian M. Scott Oct 3 '15 at 15:13
  • $\begingroup$ (1)The space of rationals, or any discrete space are examples in a vacuous fashion. Metrizable but having no connected subspace with more than one point.(2) A hedgehog space is a metrizable, connected example. (3) Any connected linear space. (4) A hedgehog-like space made of connected linear spaces instead of copies of R.......These examples all have small-inductive dimension 1 or 0. $\endgroup$ – DanielWainfleet Oct 19 '15 at 8:04

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