Definition:Finer Filter on Set/Strictly Finer
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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.
- 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