Theorems · Theorem · general topology
IsProperMap.isCompact_preimage
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
IsProperMap f → ∀ {K : Set Y}, IsCompact K → IsCompact (f ⁻¹' K)The preimage of a compact set by a proper map is again compact. See also
isProperMap_iff_isCompact_preimage which proves that this property completely characterizes
proper map when the codomain is compactly generated and Hausdorff.
- Defined in
- Mathlib.Topology.Maps.Proper.Basic
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 79 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
- Filterproof · cited by 8,121
- nhdsproof · cited by 5,554
- Set.preimagestatement and proof · cited by 4,946
- IsCompactstatement and proof · cited by 1,282
- Filter.principalproof · cited by 740
- Ultrafilterproof · cited by 193
- Ultrafilter.toFilterproof · cited by 172
- IsProperMapstatement and proof · cited by 66
- Filter.comap_principalproof · cited by 47
- Ultrafilter.mapproof · cited by 26
Cited by12
Results whose statement or proof uses this declaration.
- isProperMap_iff_isClosedMap_and_compact_fibersproof · cited by 5
- isProperMap_iff_isCompact_preimageproof · cited by 3
- ProperVAdd.isCompact_setOfPred_inter_nonemptyproof · cited by 2
- IsProperMap.restrictPreimageproof · cited by 2
- ProperSMul.isCompact_setOfPred_inter_nonemptyproof · cited by 2
- ContinuousMap.uniformSpace_eq_inf_precomp_of_coverproof · cited by 1
- IsProperMap.isSpectralMapproof · cited by 1
- properSpace_iff_isProperMap_distproof · cited by 0
- LipschitzWith.properSpaceproof · cited by 0
- IsCompact.preimage_smulproof · cited by 0
- IsCompact.preimage_vaddproof · cited by 0
- ContinuousMap.uniformSpace_eq_iInf_precomp_of_coverproof · cited by 0