Definition:Filter on Set/Trivial Filter

Definition
Let $X$ be a set. Any filter $\mathcal F$ on $X$ by definition specifically does not include the empty set $\varnothing$.

If a filter $\mathcal F$ were to include $\varnothing$, then from Empty Set is Subset of All Sets it would follow that every subset of $X$ would have to be in $\mathcal F$, and so $\mathcal F = \mathcal P \left({X}\right)$.

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