Theorems · Theorem · general topology
IsGenericPoint.mem_open_set_iff
∀ {α : Type u_1} [inst : TopologicalSpace α] {x : α} {S U : Set α},
IsGenericPoint x S → IsOpen U → (x ∈ U ↔ (S ∩ U).Nonempty)- Defined in
- Mathlib.Topology.Sober
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 80 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.
Cites8
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
- Set.Nonemptystatement and proof · cited by 2,627
- IsOpenstatement and proof · cited by 2,400
- Specializes.mem_openproof · cited by 27
- IsGenericPointstatement and proof · cited by 26
- IsGenericPoint.specializesproof · cited by 5
- IsGenericPoint.memproof · cited by 5
Cited by4
Results whose statement or proof uses this declaration.
- Topology.IsOpenEmbedding.quasiSoberproof · cited by 2
- AlgebraicGeometry.IsAffineOpen.primeIdealOf_genericPointstatement and proof · cited by 1
- AlgebraicGeometry.functionField_isFractionRing_of_isAffineOpenproof · cited by 0
- IsGenericPoint.disjoint_iffproof · cited by 0