CounterexampleConstruction: AI-generatedVerification: 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.

Assuming choice, a Lindelöf space whose square is not Lindelöf: the lower-limit line

Statement refuted

Assuming countable choice, Lindelöfness is productive.

Facts & Assumptions

Given: The lower-limit line $S$ and its square $S^{2}$ .

[L1]

The lower-limit line has Lindelöf degree $ℵ_{0}$ , hence is Lindelöf (For the lower-limit line, $χ = d = L = c = ℵ_{0}$ and $w = 2^{ℵ_{0}}$ under choice).

[L2]

The lower-limit plane $S^{2}$ is not Lindelöf (Assuming choice, the lower-limit plane is first countable, separable, and ccc, but not second countable or Lindelöf).

[L3]

The false statement being exhibited asserts that products of Lindelöf spaces are Lindelöf (Assuming countable choice, refuted: Lindelöfness is productive).

Counterexample

technique · direct

1.1

The factor $S$ is Lindelöf by [L1].

1.2

Its square $S^{2}$ is not Lindelöf by [L2].

2.1

Hence the product $S \times S$ fails the conclusion while each displayed factor satisfies the hypothesis, refuting the assertion in [L3].

step 1.1step 1.2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

24 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

UCR General Topology Notes (standard reference, not scraped)
Sorgenfrey plane (Wikipedia) (standard reference, not scraped)
Lower limit topology (Wikipedia) (standard reference, not scraped)