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Let $K$ be a non-empty compact set of reals without isolated points. Considering the usual topology, is it possible that $K$ is not totally disconnected (i.e., the singletons are not the unique connected sets) and not regular closed (i.e., the closure of the interior of $K$ is not $K$ itself)?

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How about $C\cup [1,2],$ where $C$ is the usual Cantor set?

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  • $\begingroup$ I did know I forgot something obvious, thanks anyway $\endgroup$ – Paolo Leonetti Aug 24 '17 at 19:21

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