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Paul Fabel
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Perhaps the theorem at the common intersection of the various ambiguous interpretations of the inquiry at hand is the following:

If the planar open set U contains a continuum K, if the planar complement of K is connected, then there exists a connected and simply connected open set V such that K is a subset of V, and V is a subset of U.

This follows, for example, from the fact that K is cellular, the nested intersection of closed topological planar disks. Surround K by a slightly larger disk D, so that D is contained in U, and the interior of D is the coveted open set V.

For a barehanded argument of the cellularity claim, prove by induction that if C is a planar simple closed curve, if C is the cancatanation of vertical and horizontal segments of length 1, then C bounds a topological disk.

To obtain the mentioned disk D, tile the plane by very small squares, and extract a curve C from the boundaries of the tiled squares.

Of course the argument sketched above does not use any heavy artillery such as the Jordan curve theorem or the Schoenflies theorem.

Paul Fabel
  • 2k
  • 15
  • 23