Theorems · Theorem · general topology
Topology.IsClosedEmbedding.isCompact_preimage
∀ {X : Type u} {Y : Type v} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
Topology.IsClosedEmbedding f → ∀ {K : Set Y}, IsCompact K → IsCompact (f ⁻¹' K)The preimage of a compact set under a closed embedding is a compact set.
- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- Set.preimagestatement · cited by 4,946
- IsCompactstatement and proof · cited by 1,282
- Topology.IsClosedEmbeddingstatement and proof · cited by 195
- Topology.IsClosedEmbedding.isClosed_rangeproof · cited by 41
- Topology.IsClosedEmbedding.isInducingproof · cited by 10
- Topology.IsInducing.isCompact_preimageproof · cited by 1
Cited by12
Results whose statement or proof uses this declaration.
- Topology.IsClosedEmbedding.tendsto_cocompactproof · cited by 9
- HasCompactSupport.comp_isClosedEmbeddingproof · cited by 3
- MeasureTheory.contDiffOn_convolution_right_with_paramproof · cited by 2
- Topology.IsConstructible.image_of_isClosedEmbeddingproof · cited by 1
- Topology.IsClosedEmbedding.sigmaCompactSpaceproof · cited by 1
- IsCompact.sigma_exists_finite_sigma_eqproof · cited by 1
- HasCompactMulSupport.comp_isClosedEmbeddingproof · cited by 1
- IsCompactOperator.codRestrictproof · cited by 1
- Topology.IsClosedEmbedding.weaklyLocallyCompactSpaceproof · cited by 1
- Submonoid.units_isCompactproof · cited by 0
- TopologicalSpace.NonemptyCompacts.isCompact_subsets_of_isCompactproof · cited by 0
- UpperHalfPlane.isProperMap_smul_Iproof · cited by 0