Theorems · Theorem · general topology
isHomeomorph_iff_continuous_isClosedMap_bijective
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
IsHomeomorph f ↔ Continuous f ∧ IsClosedMap f ∧ Function.Bijective fA map is a homeomorphism iff it is continuous, closed and bijective.
- Defined in
- Mathlib.Topology.Homeomorph.Lemmas
- Cited by
- 1 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.
Cites14
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
- Compl.complproof · cited by 2,925
- Continuousstatement and proof · cited by 2,592
- IsOpenproof · cited by 2,400
- Function.Bijectivestatement and proof · cited by 863
- IsClosedMapstatement and proof · cited by 138
- IsHomeomorphstatement and proof · cited by 69
- IsOpen.isClosed_complproof · cited by 50
- isClosed_compl_iffproof · cited by 35
- IsHomeomorph.bijectiveproof · cited by 9
- IsHomeomorph.continuousproof · cited by 8
Cited by1
Results whose statement or proof uses this declaration.
- isHomeomorph_iff_continuous_bijectiveproof · cited by 2