TheoremStatement: Literature-sourcedProof: 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.

Every second countable space is first countable

Statement

Every second countable topological space is first countable.

Facts & Assumptions

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

Proof

technique · direct

1.1

For $x \in X$ , the subfamily ${B \in B : x \in B}$ is countable and refines every neighbourhood of $x$ because $B$ 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 · two levels

8 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

UCR General Topology Notes (standard reference, not scraped)
Second-countable space (Wikipedia) (standard reference, not scraped)