Theorems · Theorem · algebraic geometry
AlgebraicGeometry.QuasiCompact.isCompact_preimage
∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} [self : AlgebraicGeometry.QuasiCompact f] (U : Set ↥Y),
IsOpen U → IsCompact U → IsCompact (⇑f ⁻¹' U)The preimage of a compact open set under a quasi-compact morphism between schemes is compact.
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement · cited by 53,352
- Quiver.Homstatement and proof · cited by 32,603
- Set.preimagestatement · cited by 4,946
- CategoryTheory.ConcreteCategory.homstatement · cited by 4,022
- TopCat.carrierstatement · cited by 3,184
- AlgebraicGeometry.Schemestatement and proof · cited by 2,540
- ContinuousMapstatement · cited by 2,491
- IsOpenstatement · cited by 2,400
- CommRingCatstatement · cited by 2,333
- AlgebraicGeometry.PresheafedSpace.carrierstatement · cited by 2,020
- AlgebraicGeometry.SheafedSpace.toPresheafedSpacestatement · cited by 1,988
Cited by5
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.Scheme.Hom.ker_applyproof · cited by 14
- AlgebraicGeometry.QuasiCompact.compactSpace_of_compactSpaceproof · cited by 7
- AlgebraicGeometry.exists_map_preimage_le_map_preimageproof · cited by 3
- AlgebraicGeometry.exists_isAffineOpen_preimage_eqproof · cited by 1