Definition:Principal Ultrafilter

Definition

Let $S$ be a set.

Let $\powerset S$ denote the power set of $S$.

Let $\FF \subset \powerset S$ be an ultrafilter on $S$ with a cluster point.

Then $\FF$ is a principal ultrafilter on $S$.

Nonprincipal Ultrafilter

Let $\FF \subset \powerset S$ be an ultrafilter on $S$ which does not have a cluster point.

Then $\FF$ is a nonprincipal ultrafilter on $S$.

Also known as

A principal ultrafilter is also known as a fixed ultrafilter.

Also see

• From Cluster Point of Ultrafilter is Unique, an ultrafilter is either principal or nonprincipal -- there are no other options.

• In Principal Ultrafilter is All Sets Containing Cluster Point it is shown that a principal ultrafilter on $S$ whose cluster point is $x$ is the set of all subsets of $S$ which contain $x$.

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