TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

Every second countable space is first countable

Statement

Every second countable topological space is first countable.

Facts & Assumptions

Given: A countable basis $\mathcal B$ for $X$ (Second countability: an at most countable basis for the topology).

Proof

technique · direct

1.1

For $x\in X$ , the subfamily $\{B\in\mathcal B:x\in B\}$ is countable and refines every neighbourhood of $x$ because $\mathcal B$ is a basis.

given

2.1

Thus it is a countable neighbourhood base at every $x$ , which is first countability.

step 1.1∎

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: 17 results over 8 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)
Second-countable space (Wikipedia) (standard reference, not scraped)