Definition:Finer Filter on Set/Strictly Finer

Definition

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 $\FF$.

Also known as

A strictly finer filter than $\FF$ can also be referred to as a proper superfilter of $\FF$.

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

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Filters