DefinitionDefinition: AI-adaptedProof: 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.

The tail filter of a net

Definition

For a net $x : D \to X$ , put $T_{d} = {x_{e} : d \leq e}$ and $B_{x} = {T_{d} : d \in D}$ . This is a filter base: it is nonempty, each $T_{d}$ contains $x_{d}$ , and if $f \geq d, e$ 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 $x$ :

$F_{x} = {A \subseteq X : some d \in D has T_{d} \subseteq A} .$

Thus $A \in F_{x}$ 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 · two levels

9 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)
Filter (set theory) (Wikipedia) (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)