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.

A net is eventually or frequently in a subset of its codomain

Definition

Let $x : D \to X$ be a net (Directed preorders and nets) and let $S \subseteq X$ .

$x$ is eventually in $S$ if some $d_{0} \in D$ satisfies $x_{d} \in S$ for every $d \geq d_{0}$ .
$x$ is frequently in $S$ if, for every $d_{0} \in D$ , there is $d \geq d_{0}$ with $x_{d} \in S$ .

The net is frequently in $S$ exactly when it is not eventually in $X ∖ S$ : negating the first displayed existential-universal condition gives the second one.

Depends on

Directed preorders and nets

Used by

Dependency tree · two levels

3 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)
Net (mathematics) (Wikipedia) (standard reference, not scraped)