http://openproblemgarden.org/op/a_funcoid_related_to_directed_topological_spaces Comments for "A funcoid related to directed topological spaces" en http://openproblemgarden.org/op/a_funcoid_related_to_directed_topological_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> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Let <img class="teximage" src="/files/tex/201b5ff8bf9045c34a583adc2741b00adf1fd14c.png" alt="$ R $" /> be the complete funcoid corresponding to the usual topology on extended real line <img class="teximage" src="/files/tex/3252019c60a83f00ff396d823dbff8040639f409.png" alt="$ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $" />. Let <img class="teximage" src="/files/tex/45f96d07de2ad307ec6b9d5fbad7c02d93d9eaf2.png" alt="$ \geq $" /> be the order on this set. Then <img class="teximage" src="/files/tex/5521c999ae08fc16a7a797a3fd66316435ad7aff.png" alt="$ R\sqcap^{\mathsf{FCD}}\mathord{\geq} $" /> is a complete funcoid. </div> <div class="envtheorem"><b>Proposition</b>&nbsp;&nbsp; It is easy to prove that <img class="teximage" src="/files/tex/4a22ece277f13be752937ec312efed1484d5d2b8.png" alt="$ \langle R\sqcap^{\mathsf{FCD}}\mathord{\geq}\rangle \{x\} $" /> is the infinitely small right neighborhood filter of point <img class="teximage" src="/files/tex/4e57a21194d8d5a659e259a111ed13a9c23b52a1.png" alt="$ x\in[-\infty,+\infty] $" />. </div> <p>If proved true, the conjecture then can be generalized to a wider class of posets.</p> </tr></td> </table> </td> </tr> </table> Porton, Victor Topology http://openproblemgarden.org/op/a_funcoid_related_to_directed_topological_spaces#comment Thu, 28 Jul 2016 21:36:17 +0200 porton 60019 at http://openproblemgarden.org