Theorems · Theorem · general topology
isProperMap_iff_isClosedMap_of_inj
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
Continuous f → Function.Injective f → (IsProperMap f ↔ IsClosedMap f)An injective and continuous function is proper if and only if it is closed.
- Defined in
- Mathlib.Topology.Maps.Proper.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- Continuousstatement and proof · cited by 2,592
- IsClosedMapstatement and proof · cited by 138
- IsProperMapstatement and proof · cited by 66
- Set.subsingleton_singletonproof · cited by 36
- IsProperMap.isClosedMapproof · cited by 14
- Set.Subsingleton.preimageproof · cited by 9
- isProperMap_iff_isClosedMap_and_compact_fibersproof · cited by 5
- Set.Subsingleton.isCompactproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- isProperMap_of_isClosedMap_of_injproof · cited by 3