openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /CompactQuotientProper.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.CompactDiscreteQuotient | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Topology | |
| namespace CriticalPercolation.CompactQuotient | |
| variable {A : Type*} [Group A] [TopologicalSpace A] [IsTopologicalGroup A] | |
| variable (H : Subgroup A) [H.Normal] | |
| lemma proper_mk_of_compact (hH : IsCompact (H : Set A)) : | |
| IsProperMap (QuotientGroup.mk' H) := by | |
| apply isProperMap_iff_isClosedMap_and_compact_fibers.mpr | |
| exact ⟨QuotientGroup.continuous_mk,QuotientGroup.isClosedMap_coe hH, | |
| fun q => compact_preimage_of_finite H hH {q} (Set.finite_singleton q)⟩ | |
| lemma locallyCompact_of_compact_kernel (hH : IsCompact (H : Set A)) | |
| [LocallyCompactSpace (A ⧸ H)] : LocallyCompactSpace A := by | |
| obtain ⟨K,hK,hK1⟩ := exists_compact_mem_nhds (QuotientGroup.mk' H (1 : A)) | |
| exact ((proper_mk_of_compact H hH).isCompact_preimage hK).locallyCompactSpace_of_mem_nhds_of_group | |
| (QuotientGroup.continuous_mk.continuousAt hK1) | |
| end CriticalPercolation.CompactQuotient | |
| end | |
| end | |
| end OAI | |