DefinitionDefinition: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)judge 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.

The canonical net indexed by the pairs $(A,x)$ with $A$ in a filter and $x\in A$

Definition

Let $\mathcal F$ be a filter on $X$ . Its derived-net index set is

$E_{\mathcal F}=\{(A,x):A\in\mathcal F,\ x\in A\},$

ordered by $(A,x)\preceq(B,y)$ when $B\subseteq A$ . It is a directed preorder: filters contain no empty set, and for two indices choose $z\in A\cap B$ , so $(A\cap B,z)$ is above both. The net derived from $\mathcal F$ is

$x_{(A,x)}:=x\qquad ((A,x)\in E_{\mathcal F}).$

This construction makes no arbitrary choice, because the point $x$ is included in the index.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 6 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)
Net (mathematics) (Wikipedia) (standard reference, not scraped)
Filter (set theory) (Wikipedia) (standard reference, not scraped)

The canonical net indexed by the pairs (A,x)(A,x)(A,x) with AAA in a filter and x∈Ax\in Ax∈A