Theorems · Theorem · Lie groups
IsCompact.locallyCompactSpace_of_mem_nhds_of_group
∀ {G : Type w} [inst : TopologicalSpace G] [inst_1 : Group G] [IsTopologicalGroup G] {K : Set G},
IsCompact K → ∀ {x : G}, K ∈ nhds x → LocallyCompactSpace GIf a point in a topological group has a compact neighborhood, then the group is locally compact.
- Defined in
- Mathlib.Topology.Algebra.Group.Pointwise
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Filterstatement · cited by 8,121
- Groupstatement and proof · cited by 6,238
- nhdsstatement and proof · cited by 5,554
- IsCompactstatement and proof · cited by 1,282
- inv_invproof · cited by 494
- IsTopologicalGroupstatement and proof · cited by 469
- LocallyCompactSpacestatement · cited by 324
- Continuous.continuousAtproof · cited by 297
- mul_inv_revproof · cited by 270
- inv_mul_cancel_rightproof · cited by 70
Cited by2
Results whose statement or proof uses this declaration.
- TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_groupproof · cited by 2
- eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_groupproof · cited by 1