Theorems · Theorem · general topology
TopologicalSpace.Opens.isBasis_iff_cover
∀ {α : Type u_2} [inst : TopologicalSpace α] {B : Set (TopologicalSpace.Opens α)},
TopologicalSpace.Opens.IsBasis B ↔ ∀ (U : TopologicalSpace.Opens α), ∃ Us ⊆ B, U = sSup Us- Defined in
- Mathlib.Topology.Sets.Opens
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
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
- SetLike.coeproof · cited by 8,199
- Set.ofPredproof · cited by 6,101
- Set.iUnionproof · cited by 2,483
- iSupproof · cited by 2,415
- TopologicalSpace.Opensstatement and proof · cited by 2,040
- SupSet.sSupstatement and proof · cited by 954
- TopologicalSpace.Opens.extproof · cited by 83
- le_sSupproof · cited by 79
- TopologicalSpace.Opens.isOpenproof · cited by 64
- Set.sUnion_eq_biUnionproof · cited by 51
Cited by4
Results whose statement or proof uses this declaration.
- TopologicalSpace.Opens.IsBasis.exists_finite_of_isCompactproof · cited by 3
- TopologicalSpace.Opens.IsBasis.exists_iSup_eqproof · cited by 1
- IsOpenMap.exists_opens_image_eq_of_prespectralSpaceproof · cited by 1
- AlgebraicGeometry.Scheme.Modules.exists_isOpenCover_presentationproof · cited by 1