| dbp:mathStatement
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- If is an ultrafilter on then the following are equivalent:
is fixed, or equivalently, not free, meaning
is principal, meaning
Some element of is a finite set.
Some element of is a singleton set.
is principal at some point of which means for some
does contain the Fréchet filter on
is sequential. (en)
- Every filter on a set is a subset of some ultrafilter on (en)
- Let be a prefilter on and let
Suppose is a prefilter such that
If then
* This is the analog of "if a sequence converges to then so does every subsequence."
If is a cluster point of then is a cluster point of
* This is the analog of "if is a cluster point of some subsequence, then is a cluster point of the original sequence."
if and only if for any finer prefilter there exists some even more fine prefilter such that
* This is the analog of "a sequence converges to if and only if every subsequence has a sub-subsequence that converges to "
is a cluster point of if and only if there exists some finer prefilter such that
* This is the analog of the following statement: " is a cluster point of a sequence if and only if it has a subsequence that converges to " .
* The analog for sequences is false since there is a Hausdorff topology on and a sequence in this space that clusters at but that also does not have any subsequence that converges to (en)
- If is a filter on a compact space and is the set of cluster points of then every neighborhood of belongs to
Thus a filter on a compact Hausdorff space converges if and only if it has a single cluster point. (en)
- If is a prefilter on and then
is a cluster point of if and only if is a cluster point of (en)
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