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I am looking for the answer of the following question:
Find a subset of R^2 that is path connected but is locally connected at none of its points.
There is an answer given in 2002 (see below). But I do not understand why it is not locally connected at none of the points.
And even more, it seems that any point is connected to (1,0) and (0,0) so necessarly, they are locally path connected, therefore locally connected...
Am I right!?
Thanks for the help.
From: Peter Nyikos
Date: Dec 9, 2002
Subject: Re: munkres help
In reply to "munkres help", posted by Kimberly on Oct 27, 2002:
>I need to find a subset of R^2 that is path connected but is
>locally connected at none of its points.
>Any help and/or hints is much appreciated.
>It seems to me at least one point must be locally connected.
Have two fans emanating from the two endpoints (0,0) and (1,0).
on the closed unit interval I on the x-axis. As described below,
the fans will overlap exactly in the interval I.
The fans consist of an infinite family of line segments of
slope q as q runs over the rationals in [0, 1] for the first
fan and over the rationals in [-1, 0] in the other fan.
These line segments run from (0, 0) to the line x = 1
and from (1, 0) to the line x = 0 (the y-axis) respectively.
The first fan insures local connectedness fails everywhere in the
upper half plane except perhaps at (0,0) and the second fan takes
care of that. The second fan insures local connectedness fails
everywhere in the lower half plane except perhaps at (1, 0) and
the first fan takes care of that.
One can also produce a compact example by substituting the points
of the Cantor set for the rationals in defining the first fan and
the negatives of the points of the Cantor set for the rationals
in defining the second fan.
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