Definition:Superfilter

Definition

Let $X$ be a set, and $\mathcal P \left({X}\right)$ be the power set of $X$.

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

Then $\mathcal F'$ is called a superfilter of $\mathcal F$ if $\mathcal F \subseteq \mathcal F'$.

Finer / Coarser

If $\mathcal F'$ is a superfilter of $\mathcal F$, then:

• $\mathcal F'$ is finer than $\mathcal F$
• $\mathcal F$ is coarser than $\mathcal F'$

If $\mathcal F \subset \mathcal F'$, i.e. $\mathcal F \ne \mathcal F'$, then:

• $\mathcal F'$ is strictly finer than $\mathcal F$
• $\mathcal F$ is strictly coarser than $\mathcal F'$

If $\mathcal F \subset \mathcal F'$, then it is possible to refer to $\mathcal F'$ as a proper superfilter of $\mathcal F$, but this is not advised as there exists the danger of confusing this with the concept of a proper filter.

Comparable Filters

Two filters $\mathcal F \subset \mathcal F'$ on a set $X$ are comparable iff one is finer than the other.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Filters