\begin{definition}[Definition:Filter on Set/Trivial Filter]
Let $S$ be a set.
A filter $\FF$ on $S$ by definition specifically does ''not'' include the empty set $\O$.
If a filter $\FF$ ''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 $\FF$, and so $\FF = \powerset S$.
Such a "filter" is called '''the trivial filter''' on $S$.
Category:Definitions/Filter Theory
\end{definition}