Re: connected and disconnected subsets of R^2

quasi wrote (in part):

(2) If S is a disconnected subset of R^2, then R^2-S
contains a path connected subset with more than one point.

W. Dale Hall wrote (in part):

This, I don't know.

It's possible that one of the connected graphs of certain
discontinuous solutions to the Cauchy functional equation
that F. Burton Jones constructed in "Connected and
disconnected plane sets and the functional equation
f(x) + f(y) = f(x+y)" [Bulletin of the American Mathematical
Society 48 (1942), 115-120] can be adapted. Note that if
you delete exactly one point from such a connected graph,
what's left will be a disconnected set in R^2 (a vertical
line through that point provides a disconnection). However,
I don't know if the complement of one of Jones' examples
can have no path connected subsets containing more than
one point. [I don't have a copy of Jones' paper where I'm
at right now to check to see if perhaps he says something
that would imply this is possible.]

I do know, by the way, that if R^2 - S contains no
uncountable closed sets (a stronger property than
containing no path connected subsets with more than
one point), then S must be connected. Sierpinski
showed this in 1920 [1]. In fact, Sierpinski showed
this holds even when R^2 - S has the weaker property
of being punctiform (contains no non-singleton compact
connected set).

[1] http://matwbn.icm.edu.pl/ksiazki/fm/fm1/fm112.pdf

Dave L. Renfro

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