ExampleConstruction: AI-adaptedVerification: 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 neighbourhood-indexed net in $A$ converges to each point of $\overline{A}$

Example

Let $p \in \overline{A} \subseteq X$ . The pairs $(N, a)$ with $N$ a neighbourhood of $p$ and $a \in N \cap A$ , directed by reverse inclusion of $N$ , form an index set. The net $(N, a) \mapsto a$ lies in $A$ and converges to $p$ .

Facts & Assumptions

Given: A point $p \in \overline{A}$ in a topological space $X$ .

[L1]

Every neighbourhood of $p$ meets $A$ (A point lies in the closure of a set if and only if a net in the set converges to it).

[L2]

Finite intersections of neighbourhoods of $p$ are neighbourhoods (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open).

[L3]

Net convergence means eventual membership in each neighbourhood (Convergence and cluster points of a net in a topological space).

Verification

technique · constructive

1.1

Let $E = {(N, a) : N \in N (p), a \in N \cap A}$ and order it by $(N, a) ⪯ (M, b)$ when $M \subseteq N$ .

L1construct

2.1

For two indices, [L2] and [L1] give $c \in (N \cap M) \cap A$ ; $(N \cap M, c)$ is above both. Thus $E$ is directed.

step 1.1L1L2

2.2

The net $x_{(N, a)} = a$ is eventually in every neighbourhood $N$ of $p$ , since any pair with first coordinate $N$ is a threshold. Hence $x \to p$ .

step 1.1L3

3.1

This is the asserted net in $A$ .

step 2.2discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

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)
Net (mathematics) (Wikipedia) (standard reference, not scraped)
net (nLab) (standard reference, not scraped)

A neighbourhood-indexed net in AA converges to each point of A‾A