Alphabeta Math
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.

Under choice, L(X) is a well-defined cardinal

Statement

Assuming choice, L(X) is a well-defined cardinal.

Facts & Assumptions

Given: A topological space X and the definition in Under choice, Lindelöf degree L(X) and cellularity c(X) as raw cardinal functions.

[A1]

Under choice every set has a cardinality (Cardinal (initial ordinal) and cardinality, The well-ordering theorem).

[L1]

Every nonempty set of ordinals, and hence every nonempty set of cardinals, has a least member; this is a theorem of ZF (Trichotomy and well-ordering of the ordinals).

Proof

technique · direct
1.1

By [A1], the topology τ has a cardinality κ. Every open cover is a subcover of itself and has cardinality at most κ, so κ bounds every cover's subcover size.

A1given
2.1

Let S be the set of cardinals λ≤κ such that every open cover of X has a subcover of cardinality at most λ. By Step 1.1, κ∈S, so [L1] supplies a least member of S. Any bounding cardinal larger than κ cannot be smaller than that member, and hence this least member is exactly L(X).

step 1.1L1∎

Depends on

Used by

Cited to discharge well-definedness by Under choice, Lindelöf degree L(X) and cellularity c(X) as raw cardinal functions.

Dependency tree · two levels

23 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