Theorems · Theorem · general topology
Topology.IsClosedEmbedding.tendsto_cocompact
∀ {X : Type u} {Y : Type v} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
Topology.IsClosedEmbedding f → Filter.Tendsto f (Filter.cocompact X) (Filter.cocompact Y)A closed embedding is proper, i.e., inverse images of compact sets are contained in compacts.
Moreover, the preimage of a compact set is compact, see IsClosedEmbedding.isCompact_preimage.
- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Filter.Tendstostatement · cited by 3,814
- IsCompactproof · cited by 1,282
- Topology.IsClosedEmbeddingstatement and proof · cited by 195
- Filter.cocompactstatement · cited by 141
- Filter.HasBasis.tendsto_right_iffproof · cited by 81
- Filter.hasBasis_cocompactproof · cited by 14
- IsCompact.compl_mem_cocompactproof · cited by 13
- Topology.IsClosedEmbedding.isCompact_preimageproof · cited by 12
Cited by9
Results whose statement or proof uses this declaration.
- IsClosed.tendsto_coe_cofinite_of_isDiscreteproof · cited by 4
- tendsto_norm_comp_cofinite_atTop_of_isClosedEmbeddingproof · cited by 3
- ModularGroup.tendsto_abs_re_smulproof · cited by 1
- ModularGroup.tendsto_lcRow0proof · cited by 1
- ModularGroup.tendsto_normSq_coprime_pairproof · cited by 1
- AddEquiv.isAddHaarMeasure_mapproof · cited by 1
- Topology.IsClosedEmbedding.noncompactSpaceproof · cited by 1
- MulEquiv.isHaarMeasure_mapproof · cited by 0
- tendsto_norm_comp_cofinite_atTop_of_isClosedEmbedding'proof · cited by 0