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May 3, 2018 at 9:06 comment added R. van Dobben de Bruyn @js21: I agree that your example is much more interesting. I wanted to note that the question as stated actually has trivial counterexamples as well. – Remy "minimal counterexamples" van Dobben de Bruyn.
May 3, 2018 at 8:33 comment added js21 Yes, the connectedness of $S$ is clearly a necessary condition, and I just assumed OP forgot to require that, so I gave a counterexample with $S$ connected.
May 3, 2018 at 7:26 history answered R. van Dobben de Bruyn CC BY-SA 4.0