Definition:Ultrafilter on Set/Definition 1
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Definition
Let $S$ be a set.
Let $\FF \subseteq \powerset S$ be a filter on $S$.
Then $\FF$ is an ultrafilter (on $S$) if and only if:
- there is no filter on $S$ which is strictly finer than $\FF$
or equivalently, if and only if:
- whenever $\GG$ is a filter on $S$ and $\FF \subseteq \GG$ holds, then $\FF = \GG$.
Also see
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