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Mar 10, 2010 at 16:30 history edited Anton Petrunin CC BY-SA 2.5
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Mar 8, 2010 at 15:44 comment added Anton Petrunin A connected open set in the plane has to be path-connected (that is standard fact and it is easy to prove). So you may pass to $\epsilon$-neighborhood of your set (and take its closure if you want to keep it compact).
Mar 8, 2010 at 7:53 comment added arun s Nice answer!. Could you point me to a reference to the fact: "a connected set in the plane can be approximated by path connected set".
Mar 8, 2010 at 6:48 vote accept arun s
Mar 7, 2010 at 19:31 history edited Anton Petrunin CC BY-SA 2.5
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Mar 7, 2010 at 19:11 history answered Anton Petrunin CC BY-SA 2.5