Theorems · Definition · algebraic geometry
AlgebraicGeometry.Scheme.Hom.homeomorph
{X Y : AlgebraicGeometry.Scheme} → (f : X ⟶ Y) → [CategoryTheory.IsIso f] → ↥X ≃ₜ ↥YAn isomorphism of schemes induces a homeomorphism of the underlying topological spaces.
- Defined in
- Mathlib.AlgebraicGeometry.Scheme
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CategoryTheory.IsIso
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Quiver.Homstatement and proof · cited by 32,603
- TopCat.carrierstatement · cited by 3,184
- AlgebraicGeometry.Schemestatement and proof · cited by 2,540
- CommRingCatstatement · cited by 2,333
- AlgebraicGeometry.PresheafedSpace.carrierstatement · cited by 2,020
- AlgebraicGeometry.SheafedSpace.toPresheafedSpacestatement · cited by 1,988
- AlgebraicGeometry.LocallyRingedSpace.toSheafedSpacestatement · cited by 1,892
- AlgebraicGeometry.Scheme.toLocallyRingedSpacestatement · cited by 1,734
- CategoryTheory.IsIsostatement and proof · cited by 1,156
- Homeomorphstatement · cited by 725
- CategoryTheory.asIsoproof · cited by 177
- CategoryTheory.Iso.schemeIsoToHomeoproof · cited by 0
Cited by13
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.Scheme.Hom.opensRange_of_isIsoproof · cited by 4
- AlgebraicGeometry.IsLocallyArtinian.of_topologicalKrullDim_le_zeroproof · cited by 2
- AlgebraicGeometry.Proj.awayι_preimage_basicOpenproof · cited by 1
- AlgebraicGeometry.exists_etale_isCompl_of_quasiFiniteAtproof · cited by 1
- AlgebraicGeometry.isField_of_isIntegral_of_subsingletonproof · cited by 1
- AlgebraicGeometry.noetherianSpace_of_isAffineproof · cited by 1
- AlgebraicGeometry.GeometricallyIrreducible.compproof · cited by 0
- AlgebraicGeometry.Scheme.Hom.homeomorph_applystatement · cited by 0
- AlgebraicGeometry.Scheme.Hom.homeomorph.congr_simpstatement and proof · cited by 0
- AlgebraicGeometry.GeometricallyConnected.compproof · cited by 0
- AlgebraicGeometry.isInitial_iff_isEmptyproof · cited by 0
- AlgebraicGeometry.Scheme.eq_zeroLocus_of_isClosed_of_isAffineproof · cited by 0