Nonseparating planar continuum

Importance: Medium ✭✭
Subject: Topology
Keywords: fixed point
Recomm. for undergrads: no
Posted by: porton
on: February 6th, 2011

\begin{conjecture} Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property?

A set has the fixed point property if every continuous map from it into itself has a fixed point. \end{conjecture}

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* indicates original appearance(s) of problem.

Proof; some set of disks connected by line segments

1) Any continuous map of this set to itself must traverse these line segments, and some of them will find their fixed point within one of these line segments, as any such mapping that has a submapping that maps a line segment to itself has a fixed point within that line segment. 2) For those mappings that haven't had a fixed point within one of the line segments above, must then have a submapping that maps a part of a disk to itself. This guarantees a fixed point will be found by Brouwer's fixed point theorem. *) Some of the mapping will map separate disks to each other, and there will be no fixed point in that part of the mapping. But how are the separate disks connected? Either they are connected along a line segment, in which case the fixed point must be there (see 1) or the disks are connected by a point, in which case the fixed point must be there at that point.

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