A funcoid related to directed topological spaces

Problem's Description

Conjecture Let $R$ be the complete funcoid corresponding to the usual topology on extended real line $[-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\}$. Let $\geq$ be the order on this set. Then $R\sqcap^{\mathsf{FCD}}\mathord{\geq}$ is a complete funcoid.

Proposition It is easy to prove that $\langle R\sqcap^{\mathsf{FCD}}\mathord{\geq}\rangle \{x\}$ is the infinitely small right neighborhood filter of point $x\in[-\infty,+\infty]$.

If proved true, the conjecture then can be generalized to a wider class of posets.

1. BookIs an originAlgebraic General Topology. Volume 1

   • Victor Porton

   year of publication: 2018fulltext