Definition:Finer Filter on Set/Strictly Finer

Definition
Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathcal F, \mathcal F' \subset \mathcal P \left({S}\right)$ be two filters on $S$.

Let $\mathcal F \subset \mathcal F'$, that is, $\mathcal F \subseteq \mathcal F'$ but $\mathcal F \ne \mathcal F'$.

Then $\mathcal F'$ is strictly finer than $\mathcal F$.

Also known as
A strictly finer filter than $\mathcal F$ can also be referred to as a proper superfilter of $\mathcal F$.

However, this is not encouraged, as there exists the danger of confusing this with the concept of a proper filter.

Also see

 * Definition:Finer Filter on Set


 * Definition:Coarser Filter on Set
 * Definition:Strictly Coarser Filter on Set


 * Definition:Comparable Filters on Set