Definition:Principal Ultrafilter

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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 $S$ 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


Sources