DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Convergence and cluster points of a filter on a topological space

Definition

Let $F$ be a filter on a topological space $X$ and let $p \in X$ .

$F$ converges to $p$ , written $F \to p$ , if every neighbourhood of $p$ belongs to $F$ .
$p$ is a cluster point of $F$ if $N \cap A \neq \emptyset$ for every neighbourhood $N$ of $p$ and every $A \in F$ .

The second condition says precisely that the neighbourhood filter at $p$ and $F$ have no disjoint members.

Depends on

Dependency tree · 2 levels

Sources