DefinitionDefinition: Literature-sourcedProof: AI-generatedjudge 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 net in a topological space

Definition

Let $x : D \to X$ be a net in a topological space $X$ and let $p \in X$ .

$x$ converges to $p$ , written $x_{d} \to p$ , if it is eventually in every neighbourhood of $p$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open).
$p$ is a cluster point of $x$ if $x$ is frequently in every neighbourhood of $p$ .

Convergence implies being a cluster point. If $x$ is eventually in a neighbourhood $N$ after $d_{0}$ , then for an arbitrary threshold $d$ choose a common upper bound $e \geq d, d_{0}$ ; one has $x_{e} \in N$ , so $x$ is frequently in $N$ .

Convergence and cluster points of a net in a topological space

Definition

Depends on

Dependency tree · 2 levels

Sources