CIRCULAR SETS -- old preprint on 'circular convexity' ...

[George Baloglou]

... by George Baloglou and Phil Tracy is now available in a single pdf
file at http://www.oswego.edu/~baloglou/math/circular.html and/or
http://www.oswego.edu/~baloglou/math/circular.pdf -- unpublished work
appropriate for undergraduate courses in Geometry or Topology.

baloglou(AT)oswego.edu

[Dave L. Renfro]

George Baloglou wrote:

I've only had a chance to glance at this so far, but I
thought you might be interested in knowing (if you don't
already know this) that Cantor's proof (or more accurately,
each of the finitely many steps in his proof), that deleting
a countable set from the plane leaves a connected set,
actually shows the left over set is "circular" by your
definition. See the references, especially in Dauben's
book (which, unfortunately, is slightly unintelligible
in places) I posted at

http://groups.google.com/group/sci.math/msg/49d13d079238d5b6

There's actually an easier way to prove Cantor's result
that shows polygonal connectedness (indeed, 2-segment
connectedness -- just short of being convex), but I think
the proof Cantor used was a result of the notion of
"connected" that he introduced (in another paper,
I think), which I believe is now called something
like "chain connected" or "epsilon-chain connected
for all epsilon > 0".

Dave L. Renfro