\begin{definition}[Definition:Finer Filter on Set/Strictly Finer]
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Let $\FF, \FF' \subset \powerset S$ be two filters on $S$.
Let $\FF \subset \FF'$, that is, $\FF \subseteq \FF'$ but $\FF \ne \FF'$.
Then $\FF'$ is '''strictly finer''' than $\FF$.
\end{definition}