IsTopologicalModule.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

4 declarations, 0 contain sorry, 0 declared as axioms

Topological modules

The typeclass IsTopologicalModule R M packages a topology on M for which both scalar multiplication and addition are continuous, together with basic constructions (subobjects, products) and inducing-map properties.

Imports (4)

Imported by 3 files

class IsTopologicalModule IsTopologicalModule R M states that the topology in M makes scalar multiplication and addition into continuous maps line 32 source
Source documentation
IsTopologicalModule R M states that the topology in M makes scalar multiplication and addition into continuous maps.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Ring, TopologicalSpace, AddCommGroup, Module
2 instances, coercions and auxiliary constants left out.
```
class IsTopologicalModule extends ContinuousSMul R M, ContinuousAdd M

variable [IsTopologicalModule R M]
```
theorem Topology.IsInducing.topologicalModule line 36 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Ring, TopologicalSpace, AddCommGroup, Module, IsTopologicalModule, FunLike, LinearMapClass, Topology.IsInducing
Proof uses: IsTopologicalModule.mk, Topology.IsInducing.continuousSMul, id, continuous_id, of_eq_true, Eq, HSMul.hSMul, Eq.trans, True, forall_congr, congrFun', congrArg, map_smul, RingHom.id, eq_self, implies_true, Topology.IsInducing.continuousAdd
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
instance Submodule.instIsTopologicalModuleSubtypeMem line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Ring, TopologicalSpace, AddCommGroup, Module, IsTopologicalModule, Submodule, Subtype, Membership.mem
Proof uses: Topology.IsInducing.topologicalModule, ContinuousLinearMap, RingHom.id, Submodule.subtypeL, Topology.IsInducing.subtypeVal
20 instances, coercions and auxiliary constants left out.
instance Pi.instTopologicalModule line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Ring, TopologicalSpace, AddCommGroup, Module, IsTopologicalModule
Proof uses: IsTopologicalModule.mk, ContinuousSMul.mk, ContinuousSMul.continuous_smul, ContinuousAdd.mk, continuous_add
35 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.