Bases.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 8 curated AI-written annotations and 1204 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Pinned FLT blueprint introduction

1 declarations, 0 contain sorry, 0 declared as axioms

Bases

Material destined for Mathlib.

Imports (2)

Imported by 2 files

lemma TopologicalSpace.secondCountableTopology_of_countable_cover' find home and upstream line 20 source
Source comment
find home and upstream

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, Countable, SecondCountableTopology, Topology.IsOpenEmbedding, Exists, Eq
Proof uses: Set, Set.range, Membership.mem, Homeomorph.secondCountableTopology, Homeomorph.symm, Topology.IsEmbedding.toHomeomorph, Topology.IsOpenEmbedding.toIsEmbedding, IsOpen, Topology.IsOpenEmbedding.isOpen_range, Set.iUnion, Set.univ, Set.eq_univ_of_forall, Exists.elim, Set.mem_iUnion_of_mem, Iff.mpr, Set.mem_range, TopologicalSpace.secondCountableTopology_of_countable_cover
3 instances, coercions and auxiliary constants left out.
```
lemma TopologicalSpace.secondCountableTopology_of_countable_cover' {α : Type*}
    [TopologicalSpace α] {ι : Sort*} [Countable ι] {U : ι → Type*} [∀ i, TopologicalSpace (U i)]
    [∀ (i : ι), SecondCountableTopology (U i)]
    (f : ∀ i, U i → α) (hf : ∀ i, Topology.IsOpenEmbedding (f i))
    (hc : ∀ a, ∃ (i : ι) (u : U i), f i u = a) : SecondCountableTopology α :=
…
```