http://openproblemgarden.org/op/nonseparating_planar_continuum Comments for "Nonseparating planar continuum" en http://openproblemgarden.org/op/nonseparating_planar_continuum#comment-6912 <p>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.</p> Wed, 16 Feb 2011 09:25:13 +0100 Comet comment 6912 at http://openproblemgarden.org http://openproblemgarden.org/op/nonseparating_planar_continuum <table cellspacing="10"> <tr> <td> Author(s): <a href="/kpz_equation_central_limit_theorem"></a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/topology">Topology</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property?</p> <p>A set has the fixed point property if every continuous map from it into itself has a fixed point. </div> </tr></td> </table> </td> </tr> </table> fixed point Topology http://openproblemgarden.org/op/nonseparating_planar_continuum#comment Sun, 06 Feb 2011 19:29:32 +0100 porton 37295 at http://openproblemgarden.org