Definition:Principal Ultrafilter/Nonprincipal
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Definition
Let $S$ be a set.
Let $\powerset S$ denote the power set of $S$.
Let $\FF \subset \powerset S$ be an ultrafilter on $S$ which does not have a cluster point.
Then $\FF$ is a nonprincipal ultrafilter on $S$.
Also known as
A nonprincipal ultrafilter is also known as a free ultrafilter.
Also see
- From Cluster Point of Ultrafilter is Unique, an ultrafilter is either principal or nonprincipal -- there are no other options.
- Existence of Nonprincipal Ultrafilter
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