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 tail filter of a net

Definition

For a net $x:D\to X$ , put $T_d=\{x_e:d\le e\}$ and $\mathcal B_x=\{T_d:d\in D\}$ . This is a filter base: it is nonempty, each $T_d$ contains $x_d$ , and if $f\ge d,e$ then $T_f\subseteq T_d\cap T_e$ . Its generated filter The upward closure of a filter base is the smallest filter containing it is the tail filter of $x$ :

$\mathcal F_x=\{A\subseteq X:\text{some }d\in D\text{ has }T_d\subseteq A\}.$

Thus $A\in\mathcal F_x$ exactly when the net is eventually in $A$ . The preceding filter-base verification makes this a well-defined filter in the sense of Filter on a set.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 8 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)
Filter (set theory) (Wikipedia) (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)