LemmaStatement: 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.

Every cluster point of a universal net is a limit of that net

Statement

Every cluster point of a universal net is a limit of that net.

Facts & Assumptions

Given: A universal net $x : D \to X$ and a cluster point $p$ .

[A1]

A universal net is eventually in $S$ or eventually in $X ∖ S$ for every subset $S$ (Universal net: eventually in every subset or eventually in its complement).

[A2]

Clusterhood is frequent membership in every neighbourhood, while convergence is eventual membership in every neighbourhood (Convergence and cluster points of a net in a topological space).

Proof

technique · contradiction

1.1

Assume for a contradiction that $x$ does not converge to $p$ . Then some neighbourhood $N$ of $p$ is not an eventual set for $x$ .

A2assume-contra

2.1

By universality, $x$ is eventually in $X ∖ N$ . This contradicts frequent membership in $N$ , since an index after both thresholds would lie in $N \cap (X ∖ N)$ .

step 1.1A1A2

3.1

Therefore $x$ converges to $p$ .

step 2.1discharge-contradiction∎

Depends on

Used by

Dependency tree · two levels

5 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)