Alphabeta Math
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 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∈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.

L1
1.2

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

L1
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 · two levels

19 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