Theorems · Theorem · general topology
Topology.IsCoinducing.isOpen_preimage
∀ {X : Type u_1} {Y : Type u_2} {f : X → Y} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y],
Topology.IsCoinducing f → ∀ {s : Set Y}, IsOpen (f ⁻¹' s) ↔ IsOpen s- Defined in
- Mathlib.Topology.Maps.Basic
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- IsOpenstatement · cited by 2,400
- Topology.IsCoinducingstatement and proof · cited by 31
- Topology.isCoinducing_iffproof · cited by 3
Cited by15
Results whose statement or proof uses this declaration.
- Homeomorph.isOpen_preimageproof · cited by 7
- Topology.IsCoinducing.continuousproof · cited by 6
- SeparationQuotient.isOpenMap_mkproof · cited by 5
- AddMonoidHom.isOpenQuotientMap_of_isQuotientMapproof · cited by 3
- Topology.IsCoinducing.of_comp_iffproof · cited by 3
- coinduced_eq_induced_of_isOpenQuotientMap_of_isInducingproof · cited by 2
- Topology.IsGeneratedBy.isOpen_iffproof · cited by 2
- Topology.IsCoinducing.restrictPreimage_of_isOpenproof · cited by 1
- MonoidHom.isOpenQuotientMap_of_isQuotientMapproof · cited by 1
- AlgebraicGeometry.Flat.isQuotientMap_of_surjectiveproof · cited by 1
- Topology.IsCoinducing.locallyConnectedSpaceproof · cited by 1
- Topology.IsQuotientMap.isClopen_preimageproof · cited by 0