Exercise
13
[29J]Prerequisites:[06M].Difficulty:*.(Proposed on 2022-11-24) Let \(x_ 0\in \overline{{\mathbb {R}}}\) and \(\mathcal F\) all the neighbourhoods of \(x_ 0\). We associate the ordering
\[ I,J\in {\mathcal F}~ ~ , I\le J \iff I\supseteq J \]
show that this is a filtering ordering.
(This holds both for “deleted” and for “full” neighbourhoods; for ‘left”, “right”, or “bilateral” neighbourhoods).
(See also [0GQ] for the similar statement in topological spaces).