Definition:Filter on Set/Trivial Filter

Definition
Let $S$ be a set.

A filter $\mathcal F$ on $S$ by definition specifically does not include the empty set $\O$.

If a filter $\mathcal F$ were to include $\O$, then from Empty Set is Subset of All Sets it would follow that every subset of $S$ would have to be in $\mathcal F$, and so $\mathcal F = \powerset S$.

Such a "filter" is called the trivial filter on $S$.