Theorems · Inductive type · order theory
Topology.IsLowerSet
(α : Type u_4) → [t : TopologicalSpace α] → [Preorder α] → Prop
The lower set topology is the topology where the open sets are the lower sets. In general the lower set topology does not coincide with the lower topology.
- Cited by
- 17 results in Mathlib
- Foundations
- Depth 1 from the axioms · uses no axioms
- Assumes
- TopologicalSpacePreorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites2
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement · cited by 24,529
- Preorderstatement · cited by 7,952
Cited by20
Results whose statement or proof uses this declaration.
- Topology.IsLowerSet.closure_singletonstatement and proof · cited by 2
- Topology.IsLowerSet.isOpen_iff_isLowerSetstatement and proof · cited by 2
- Topology.IsLowerSet.nhdsKer_eq_lowerClosurestatement and proof · cited by 2
- Topology.isUpperSet_orderDualstatement · cited by 1
- Topology.IsLowerSet.closure_eq_upperClosurestatement and proof · cited by 1
- Topology.IsLowerSet.nhdsKer_singletonstatement and proof · cited by 1
- Topology.IsLowerSet.nhds_eq_principal_Iicstatement and proof · cited by 1
- Topology.IsLowerSet.topology_eqstatement and proof · cited by 1
- Topology.IsLowerSet.topology_eq_lowerSetTopologystatement and proof · cited by 1
- Topology.isLowerSet_iff_nhdsstatement and proof · cited by 0
- Topology.isLowerSet_orderDualstatement · cited by 0
- Topology.IsLowerSet.WithLowerSetHomeomorphstatement and proof · cited by 0