Theorems · Theorem · general topology
isProperMap_iff_isClosedMap_and_compact_fibers
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
IsProperMap f ↔ Continuous f ∧ IsClosedMap f ∧ ∀ (y : Y), IsCompact (f ⁻¹' {y})A map is proper if and only if it is closed and its fibers are compact.
- Defined in
- Mathlib.Topology.Maps.Proper.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Filterproof · cited by 8,121
- Set.preimagestatement and proof · cited by 4,946
- Continuousstatement and proof · cited by 2,592
- IsCompactstatement and proof · cited by 1,282
- closureproof · cited by 1,254
- Filter.principalproof · cited by 740
- inf_le_leftproof · cited by 286
- inf_le_rightproof · cited by 238
- ClusterPtproof · cited by 138
- IsClosedMapstatement and proof · cited by 138
Cited by5
Results whose statement or proof uses this declaration.
- isProperMap_iff_isCompact_preimageproof · cited by 3
- IsProperMap.restrictPreimageproof · cited by 2
- isProperMap_const_iffproof · cited by 1
- isProperMap_iff_isClosedMap_of_injproof · cited by 1
- AlgebraicGeometry.Scheme.Hom.isProperMapproof · cited by 0