Theorems · Theorem · general topology
Specializes.mem_open
∀ {X : Type u_1} [inst : TopologicalSpace X] {x y : X} {s : Set X}, x ⤳ y → IsOpen s → y ∈ s → x ∈ s- Defined in
- Mathlib.Topology.Inseparable
- Cited by
- 27 results in Mathlib
- Foundations
- Depth 76 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.
Cites5
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
- IsOpenstatement and proof · cited by 2,400
- Specializesstatement and proof · cited by 176
- specializes_iff_forall_openproof · cited by 2
Cited by27
Results whose statement or proof uses this declaration.
- TopCat.Presheaf.germ_stalkSpecializesstatement · cited by 10
- TopCat.Presheaf.stalkSpecializes_reflproof · cited by 8
- AlgebraicGeometry.Scheme.SpecMap_stalkSpecializes_fromSpecStalkproof · cited by 6
- IsOpen.stableUnderGeneralizationproof · cited by 6
- TopCat.Presheaf.germ_stalkSpecializes_assocstatement and proof · cited by 6
- IsGenericPoint.mem_open_set_iffproof · cited by 4
- AlgebraicGeometry.StructureSheaf.toStalk_stalkSpecializesproof · cited by 2
- TopCat.Presheaf.stalkSpecializes_compproof · cited by 2
- TopCat.Presheaf.stalkSpecializes_stalkFunctor_mapproof · cited by 2
- TopCat.Presheaf.stalkSpecializes_stalkPushforwardproof · cited by 2
- IsLocalRing.closed_point_mem_iffproof · cited by 1
- AlgebraicGeometry.spread_out_of_isGermInjectiveproof · cited by 1