Theorems · Theorem · general topology
OrderTopology.topology_eq_generate_intervals
∀ {α : Type u_1} {t : TopologicalSpace α} {inst : Preorder α} [self : OrderTopology α], t = Preorder.topology αThe topology is generated by open intervals Set.Ioi _ and Set.Iio _.
- Defined in
- Mathlib.Topology.Order.Basic
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 9 from the axioms · uses no axioms
- Assumes
- OrderTopology
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- Preorderstatement and proof · cited by 7,952
- OrderTopologystatement and proof · cited by 1,355
- Preorder.topologystatement · cited by 24
Cited by14
Results whose statement or proof uses this declaration.
- leftLim_eq_of_eq_botproof · cited by 11
- StrictMono.isEmbedding_of_ordConnectedproof · cited by 9
- nhds_eq_orderproof · cited by 9
- borel_eq_generateFrom_Iioproof · cited by 6
- OrderIso.continuousproof · cited by 6
- tendsto_leftLim_of_tendstoproof · cited by 5
- leftLim_eq_of_tendstoproof · cited by 5
- MeasureTheory.Filtration.rightCont_applyproof · cited by 4
- OrderTopology.continuous_iffproof · cited by 3
- leftLim_eq_of_not_tendstoproof · cited by 3
- LeftOrdContinuous.continuousWithinAt_Iicproof · cited by 2
- Dense.topology_eq_generateFromproof · cited by 1