![](/files/happy5.png)
Conjecture Let
be the complete funcoid corresponding to the usual topology on extended real line
. Let
be the order on this set. Then
is a complete funcoid.
![$ R $](/files/tex/201b5ff8bf9045c34a583adc2741b00adf1fd14c.png)
![$ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $](/files/tex/3252019c60a83f00ff396d823dbff8040639f409.png)
![$ \geq $](/files/tex/45f96d07de2ad307ec6b9d5fbad7c02d93d9eaf2.png)
![$ R\sqcap^{\mathsf{FCD}}\mathord{\geq} $](/files/tex/5521c999ae08fc16a7a797a3fd66316435ad7aff.png)
Proposition It is easy to prove that
is the infinitely small right neighborhood filter of point
.
![$ \langle R\sqcap^{\mathsf{FCD}}\mathord{\geq}\rangle \{x\} $](/files/tex/4a22ece277f13be752937ec312efed1484d5d2b8.png)
![$ x\in[-\infty,+\infty] $](/files/tex/4e57a21194d8d5a659e259a111ed13a9c23b52a1.png)
If proved true, the conjecture then can be generalized to a wider class of posets.
See Algebraic General Topology for definitions of used concepts.
Bibliography
* indicates original appearance(s) of problem.