Theorems · Theorem · general topology
isOpen_iff_generate_intervals
∀ {α : Type u} [ts : TopologicalSpace α] [inst : Preorder α] [t : OrderTopology α] {s : Set α},
IsOpen s ↔ TopologicalSpace.GenerateOpen {s | ∃ a, s = Set.Ioi a ∨ s = Set.Iio a} s- Defined in
- Mathlib.Topology.Order.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 10 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- Preorderstatement and proof · cited by 7,952
- Set.ofPredstatement and proof · cited by 6,101
- IsOpenstatement and proof · cited by 2,400
- Set.Ioistatement and proof · cited by 1,463
- OrderTopologystatement and proof · cited by 1,355
- Set.Iiostatement and proof · cited by 1,166
- OrderTopology.topology_eq_generate_intervalsproof · cited by 14
- TopologicalSpace.GenerateOpenstatement and proof · cited by 12
Cited by1
Results whose statement or proof uses this declaration.
- isOpen_lt'proof · cited by 8