TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge 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 universal exactly when its tail filter is an ultrafilter, and the canonical net of an ultrafilter is universal

Statement

A net is universal if and only if its tail filter is an ultrafilter. Moreover, the net derived from an ultrafilter is universal.

Facts & Assumptions

Given: A net $x$ in $X$ and a filter $U$ on $X$ .

[A1]

$S$ belongs to the tail filter of $x$ exactly when $x$ is eventually in $S$ (The tail filter of a net).

[A2]

A filter is an ultrafilter exactly when, for every $S \subseteq X$ , it contains $S$ or $X ∖ S$ (Characterisation of ultrafilters: every set or its complement).

[A3]

The derived net of $U$ is indexed by $(A, a)$ and later indices have first coordinate contained in $A$ (The canonical net indexed by the pairs $(A, x)$ with $A$ in a filter and $x \in A$ ).

Proof

technique · direct

1.1

By [A1], universality of $x$ says exactly that its tail filter contains $S$ or $X ∖ S$ for every $S \subseteq X$ . By [A2], this is exactly ultrafilterhood.

A1A2

1.2

If $U$ is an ultrafilter and $S \subseteq X$ , [A2] gives $S \in U$ or $X ∖ S \in U$ . In the first case an index $(S, a)$ exists and every later value lies in $S$ by [A3]; the second case is identical.

A2A3

2.1

Thus the derived net of an ultrafilter is universal, completing both assertions.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)
Filter (set theory) (Wikipedia) (standard reference, not scraped)
Ultrafilter (Wikipedia) (standard reference, not scraped)