CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Assuming choice, a separable space with a nonseparable subspace: the lower-limit plane and its antidiagonal

Statement refuted

Separability is hereditary.

Facts & Assumptions

Given: The lower-limit plane $P$ and its antidiagonal $A=\{(x,-x):x\in\mathbb R\}$ .

[L1]

The lower-limit plane is separable, and its antidiagonal is an uncountable discrete subspace (Assuming choice, the lower-limit plane is first countable, separable, and ccc, but not second countable or Lindelöf).

[L2]

The false statement being exhibited asserts that every subspace of a separable space is separable (Refuted: separability is hereditary).

Counterexample

technique · direct

1.1

By [L1], the space $P$ is separable.

1.2

By [L1], the subspace $A$ is uncountable and discrete, so it has no at most countable dense subset.

2.1

Thus $P$ is a separable space with the nonseparable subspace $A$ , contradicting the assertion recalled in [L2].

step 1.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 86 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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