http://openproblemgarden.org/op/generalized_path_connectedness_in_proximity_spaces Comments for "Generalized path-connectedness in proximity spaces" en http://openproblemgarden.org/op/generalized_path_connectedness_in_proximity_spaces#comment-75326 <p><a href="http://math.stackexchange.com/questions/691643/a-lemma-to-solve-a-conjecture-about-connectedness-in-proximity-spaces" title="http://math.stackexchange.com/questions/691643/a-lemma-to-solve-a-conjecture-about-connectedness-in-proximity-spaces">http://math.stackexchange.com/questions/691643/a-lemma-to-solve-a-conjec...</a></p> <p>--<br /> Victor Porton - <a href="http://www.mathematics21.org" title="http://www.mathematics21.org">http://www.mathematics21.org</a></p> Wed, 26 Feb 2014 20:44:24 +0100 porton comment 75326 at http://openproblemgarden.org http://openproblemgarden.org/op/generalized_path_connectedness_in_proximity_spaces <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/porton_victor">Porton</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> <p>Let <img class="teximage" src="/files/tex/6468f8f4e0c9f7a9784077872a149582dd4e62fc.png" alt="$ \delta $" /> be a <a href="http://en.wikipedia.org/wiki/Proximity_space">proximity</a>.</p> <p>A set <img class="teximage" src="/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png" alt="$ A $" /> is connected regarding <img class="teximage" src="/files/tex/6468f8f4e0c9f7a9784077872a149582dd4e62fc.png" alt="$ \delta $" /> iff <img class="teximage" src="/files/tex/979884275fcadce3909238d1ad26696960c676eb.png" alt="$ \forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right) $" />.</p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; The following statements are equivalent for every endofuncoid <img class="teximage" src="/files/tex/12e00f6f7e80e7b1fd1b89a31a7e0abe4c1b1302.png" alt="$ \mu $" /> and a set <img class="teximage" src="/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png" alt="$ U $" />:<br /> <ol class="enumerate"> \item <img class="teximage" src="/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png" alt="$ U $" /> is connected regarding <img class="teximage" src="/files/tex/12e00f6f7e80e7b1fd1b89a31a7e0abe4c1b1302.png" alt="$ \mu $" />. \item For every <img class="teximage" src="/files/tex/b9afee507ee1fdd9052fada0ecf84737e5f57da8.png" alt="$ a, b \in U $" /> there exists a totally ordered set <img class="teximage" src="/files/tex/6bd29b775b103849d9f752b5b7bcb156d949341f.png" alt="$ P \subseteq U $" /> such that <img class="teximage" src="/files/tex/823bbdacb0b048c6f8f9e75c63e39794dd918e62.png" alt="$ \min P = a $" />, <img class="teximage" src="/files/tex/3def450ce9e9804c9d4a8be3ece00739a3893a85.png" alt="$ \max P = b $" />, and for every partion <img class="teximage" src="/files/tex/69a8a2715ddfa2480ba4a8321219c905c5402338.png" alt="$ \{ X, Y \} $" /> of <img class="teximage" src="/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png" alt="$ P $" /> into two sets <img class="teximage" src="/files/tex/302cdeba125e821f3406302c9789229d48f42ea7.png" alt="$ X $" />, <img class="teximage" src="/files/tex/6e8160788c99301d68bd6cf12fcc0ed07fd138d7.png" alt="$ Y $" /> such that <img class="teximage" src="/files/tex/4b87461ca127a265ebe9b2b7dcf0ccbed211c81e.png" alt="$ \forall x \in X, y \in Y : x &lt; y $" />, we have <img class="teximage" src="/files/tex/0ef560be389646efd1fdde5ebc9afc9ac98ee64e.png" alt="$ X \mathrel{[ \mu]^{\ast}} Y $" />. </ol> </div> </tr></td> </table> </td> </tr> </table> Porton, Victor connected connectedness proximity space Topology http://openproblemgarden.org/op/generalized_path_connectedness_in_proximity_spaces#comment Sat, 01 Feb 2014 21:55:50 +0100 porton 59896 at http://openproblemgarden.org