Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Convergence and cluster points of a filter on a topological space
Definition
Let be a filter on a topological space and let .
- converges to , written , if every neighbourhood of belongs to .
- is a cluster point of if for every neighbourhood of and every .
The second condition says precisely that the neighbourhood filter at and have no disjoint members.
Depends on
Used by
- Complete uniform space: every Cauchy filter converges Definition
- Assuming the ultrafilter lemma, a free ultrafilter on ℕ converges to the added point in the one-point convergent-sequence space Example
- A Cauchy filter with a cluster point converges to that point Lemma
- A continuous map of compact Hausdorff spaces is an ultrafilter-algebra homomorphism Lemma
- A filter and its canonical derived net have the same limits and cluster points Lemma
- A given ultrafilter on a compact Hausdorff space has a unique limit Lemma
- A net and its tail filter have the same limits and cluster points Lemma
- Every cluster point of an ultrafilter is a limit of that ultrafilter Lemma
- Every convergent filter on a uniform space is Cauchy Lemma
- Under the ultrafilter lemma, an ultrafilter algebra maps each ultrafilter to its unique limit Lemma
- Assuming the ultrafilter lemma, compactness is equivalent to every net having a cluster point, every net having a convergent subnet, every filter having a cluster point, and every ultrafilter converging Theorem
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- WVU Math 581 Topology I (standard reference, not scraped)
- Filter (set theory) (Wikipedia) (standard reference, not scraped)
- ultrafilter (nLab) (standard reference, not scraped)