CorollaryStatement: Literature-sourcedProof: 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.

Under choice, the five cardinal functions recover first countability, second countability, separability, Lindelöfness, and ccc at the $\aleph_0$ threshold

Statement

Assuming choice, $X$ is first countable iff $\chi(X)\le\aleph_0$ , second countable iff $w(X)\le\aleph_0$ , separable iff $d(X)\le\aleph_0$ , Lindelöf iff $L(X)\le\aleph_0$ , and ccc iff $c(X)\le\aleph_0$ .

Facts & Assumptions

Given: A topological space $X$ and the Axiom of Choice, with the five raw cardinal functions and the named countability properties.

[L1]

The minima defining $w(X)$ , $d(X)$ , $\chi(x,X)$ and $L(X)$ , and the suprema defining $\chi(X)$ and $c(X)$ , exist as cardinals (Under choice, $w(X)$ is a well-defined cardinal, Under choice, $d(X)$ is a well-defined cardinal, Under choice, $\chi(x,X)$ and $\chi(X)$ are well-defined cardinals, Under choice, $L(X)$ is a well-defined cardinal, Under choice, $c(X)$ is a well-defined cardinal).

[L2]

First countability means a countable local base at every point, second countability means a countable basis, separability means a countable dense subset, ccc means that every pairwise-disjoint family of nonempty open sets is countable, and Lindelöfness means that every open cover has a countable subcover (First countable space: a countable neighbourhood base at every point, Second countability: an at most countable basis for the topology, Separability: the existence of an at most countable dense subset, The countable chain condition: every pairwise-disjoint family of nonempty open sets is at most countable, Countably compact, Lindel"of, sequentially compact, limit point compact and $\sigma$ -compact spaces, and relatively compact subsets).

Proof

technique · direct

1.1

By [L1], $w(X)$ and $d(X)$ are the least cardinalities of a basis and a dense subset, respectively; hence $w(X)\le\aleph_0$ and $d(X)\le\aleph_0$ say exactly that such a basis and such a dense subset are at most countable.

1.2

By [L1], $L(X)\le\aleph_0$ says that every open cover has a subcover of at most countable cardinality, and $c(X)\le\aleph_0$ says that every pairwise-disjoint family of nonempty open sets is at most countable.

1.3

Since $\chi(X)=\sup\{\chi(x,X):x\in X\}$ , one has $\chi(X)\le\aleph_0$ exactly when every point has a local base of cardinality at most $\aleph_0$ .

2.1

The descriptions in steps 1.1, 1.2 and 1.3 are precisely the definitions in [L2], so they yield the five asserted equivalences.

step 1.1step 1.2step 1.3L2∎

Depends on

Used by

Implication, preservation, counterexample, and choice ledger for the countability axioms Remark

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 93 results over 18 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

D. H. Fremlin, Measure Theory, Chapter 5A (standard reference, not scraped)

Under choice, the five cardinal functions recover first countability, second countability, separability, Lindelöfness, and ccc at the ℵ0\aleph_0ℵ0​ threshold