Theorems · Theorem · algebraic geometry
AlgebraicGeometry.isCompact_iff_finite_and_eq_biUnion_affineOpens
∀ {X : AlgebraicGeometry.Scheme} {U : X.Opens}, IsCompact ↑U ↔ ∃ s, s.Finite ∧ U = ⨆ i ∈ s, ↑i- Cited by
- 6 results in Mathlib
- Foundations
- Depth 148 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites28
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
- SetLike.coestatement and proof · cited by 8,199
- Set.Elemstatement and proof · cited by 7,166
- TopCat.carrierstatement and proof · cited by 3,184
- AlgebraicGeometry.Schemestatement and proof · cited by 2,540
- Set.iUnionproof · cited by 2,483
- iSupstatement and proof · cited by 2,415
- IsOpenproof · cited by 2,400
- CommRingCatstatement · cited by 2,333
- TopologicalSpace.Opensproof · cited by 2,040
- AlgebraicGeometry.PresheafedSpace.carrierstatement and proof · cited by 2,020
- AlgebraicGeometry.SheafedSpace.toPresheafedSpacestatement and proof · cited by 1,988
Cited by6
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.compact_open_induction_onproof · cited by 2
- AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_rightproof · cited by 2
- AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_rightproof · cited by 1
- AlgebraicGeometry.isCompact_basicOpenproof · cited by 1