Its well known that the removal of countably many points from the euclidean plane doesn't affect path-connectedness.I am interested to know that if I remove a countable collection of pairwise disjoint closed balls from the plane,will it remain path connected or not.Any help would be highly appreciated,thanks in advance!
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Remove the closed balls of radius $0.5$ centered at, say, the even integers on the $x$ axis in $\mathbb{R}^2$. Do you still have path-connectedness? – SystematicDisintegration Jun 10 '18 at 10:04
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@SystematicDisintegration A single example where pathconnectedness is preserved won't solve the general problem – Hagen von Eitzen Jun 10 '18 at 10:06
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@Hagen von Eitzen That is true. I merely wished to produce a simple example where pathconnectedness holds. – SystematicDisintegration Jun 10 '18 at 10:07