Dodging Holes in the Plane

[William Elliot]

If the union of a countable pairwise disjoint collection of
bounded, simply connected open sets is removed from the real
plain, is the result path connected?

If the union of a countable pairwise disjoint collection of
simply connected compact sets with non-empty interior, is
removed from the real plain, is the result path connected?

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[Bill]

I don't know the answers, but I would see if I could use a tool like
the
function y=sin(1/x), to achieve a "negative" result.   "Fun" problem!  
-Bill