Theorems · Theorem · general topology
Topology.IsGeneratedBy.isOpen_iff
∀ {ι : Type t} (X : ι → Type u) [inst : (i : ι) → TopologicalSpace (X i)] {Y : Type v} [tY : TopologicalSpace Y]
[Topology.IsGeneratedBy X Y] {U : Set Y}, IsOpen U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsOpen (⇑f ⁻¹' U)- Defined in
- Mathlib.Topology.Convenient.GeneratedBy
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.preimagestatement and proof · cited by 4,946
- ContinuousMapstatement and proof · cited by 2,491
- IsOpenstatement and proof · cited by 2,400
- Topology.IsGeneratedBystatement and proof · cited by 38
- Topology.IsQuotientMap.isCoinducingproof · cited by 35
- Topology.WithGeneratedByTopology.equivproof · cited by 24
- Homeomorph.isQuotientMapproof · cited by 22
- Topology.IsCoinducing.isOpen_preimageproof · cited by 15
- Topology.IsGeneratedBy.homeomorphproof · cited by 6
Cited by2
Results whose statement or proof uses this declaration.
- IsOpen.isGeneratedByproof · cited by 1
- DeltaGeneratedSpace.isOpen_iffproof · cited by 0