Theorems · Theorem · general topology
isOpen_iUnion
∀ {X : Type u} {ι : Sort v} [inst : TopologicalSpace X] {f : ι → Set X}, (∀ (i : ι), IsOpen (f i)) → IsOpen (⋃ i, f i)- Defined in
- Mathlib.Topology.Basic
- Cited by
- 88 results in Mathlib
- Foundations
- Depth 11 from the axioms · uses propext, Quot.sound
- Assumes
- TopologicalSpace
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.iUnionstatement · cited by 2,483
- IsOpenstatement and proof · cited by 2,400
- Set.forall_mem_rangeproof · cited by 135
- isOpen_sUnionproof · cited by 15
Cited by88
Results whose statement or proof uses this declaration.
- TopologicalSpace.Opens.iSup_mkstatement · cited by 41
- isOpen_biUnionproof · cited by 35
- IsOpen.unionproof · cited by 19
- AlgebraicGeometry.isCompact_iff_finite_and_eq_biUnion_affineOpensproof · cited by 6
- MeasureTheory.Content.innerContent_iUnion_natstatement · cited by 6
- MeasureTheory.Content.outerMeasure_opensproof · cited by 6
- AlgebraicGeometry.IsZariskiLocalAtTarget.of_iSup_eq_topproof · cited by 5
- TopologicalSpace.NoetherianSpace.isCompactproof · cited by 5
- TopologicalSpace.Opens.iSup_defstatement · cited by 5
- TopologicalSpace.IsOpenCover.comapproof · cited by 4
- TopologicalSpace.IsTopologicalBasis.isOpenMap_iffproof · cited by 4
- AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE_locallyproof · cited by 3