Theorems · Theorem · general topology
semicontinuous_restrict_iff
∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] {r : α → β → Prop} {s : Set α},
Semicontinuous (s.domRestrict r) ↔ SemicontinuousOn r s- Defined in
- Mathlib.Topology.Semicontinuity.Defs
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 73 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.
Cites11
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.Elemstatement and proof · cited by 7,166
- nhdsproof · cited by 5,554
- Filter.Eventuallyproof · cited by 3,134
- Set.domRestrictstatement and proof · cited by 383
- Semicontinuousstatement and proof · cited by 12
- SemicontinuousWithinAtproof · cited by 12
- SemicontinuousOnstatement · cited by 10
- SetCoe.forallproof · cited by 10
- nhdsWithin_eq_map_subtype_coeproof · cited by 5
Cited by4
Results whose statement or proof uses this declaration.
- lowerSemicontinuous_restrict_iffproof · cited by 2
- lowerHemicontinuous_restrict_iffproof · cited by 0
- hasOpenLowerSections_restrict_iffproof · cited by 0
- upperHemicontinuousOn_iff_restrictproof · cited by 0