LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

How 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.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Every cluster point of an ultrafilter is a limit of that ultrafilter

Statement

Every cluster point of an ultrafilter is a limit of that ultrafilter.

Facts & Assumptions

Given: An ultrafilter $U$ on $X$ and a cluster point $p$ of it.

[A1]

$U \to p$ means every neighbourhood of $p$ belongs to $U$ , while clusterhood means every such neighbourhood meets every member of $U$ (Convergence and cluster points of a filter on a topological space).

[A2]

For every subset $S$ , an ultrafilter contains $S$ or its complement (Characterisation of ultrafilters: every set or its complement).

Proof

technique · contradiction

1.1

Assume for a contradiction that $U$ does not converge to $p$ . Then some neighbourhood $N$ of $p$ is not in $U$ .

A1assume-contra

2.1

By [A2], $X ∖ N \in U$ . But $N$ must meet every member of $U$ by clusterhood, whereas $N \cap (X ∖ N) = \emptyset$ .

step 1.1A1A2

3.1

This contradiction proves $U \to p$ .

step 2.1discharge-contradiction∎

Depends on

Used by

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 19 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

WVU Math 581 Topology I (standard reference, not scraped)
Ultrafilter (Wikipedia) (standard reference, not scraped)
Filter (set theory) (Wikipedia) (standard reference, not scraped)
ultrafilter (nLab) (standard reference, not scraped)