retract..., why?

From: boyandshark (amitgandhi_at_gmail.com)
Date: 02/28/05 

Date: 28 Feb 2005 10:58:56 -0800

Hi - I am trying to formalize a suggestion I recieved about how to show
a certain set is a retract. My situation is this:

"I have a compact (closed and bounded) subset A of R^2 that is
connected. This subset A has the following property that I call
restricted convexity: if two points a and b in A differ from one
another in at most one coordinate, then any convex combination of a and
b is also in A. Does A have the fixed point property?"

Here is the suggestion I recieved for showing why A has the FPP:

"I believe that A has the fixed point property in R^2. Here is my
reasoning. Let R=[a,b] x [c,d] be the smallest rectangle in R^2 with
horizontal and vertical sides that contains A. Let L be the lower
boundary of A in R. The set L is a continuum with only two non cut
points. Therefore L is homeomorphic to an arc. Then the points in R
below A can be retracted onto L. A similar argument can be applied to
the points of R above A. So A is a retract of R and hence has the
fixed point property."

I was hoping someone on this board might be able to elucidate this
suggestion for me a bit - for instance: why, based on the assumptions
on A, is the lower boundary L of A in R a continuum that has only two
non cut points?

Thanks in advanced for any time and consideration given to this
problem.

Amit