Theorems · Theorem · general topology
upperSemicontinuousOn_iff_isClosed_hypograph
∀ {α : Type u_1} [inst : TopologicalSpace α] {s : Set α} {γ : Type u_4} [inst_1 : LinearOrder γ]
[inst_2 : TopologicalSpace γ] [ClosedIicTopology γ] {f : α → γ},
IsClosed s → (UpperSemicontinuousOn f s ↔ IsClosed {p | p.1 ∈ s ∧ p.2 ≤ f p.1})- Defined in
- Mathlib.Topology.Semicontinuity.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
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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
- LinearOrderstatement and proof · cited by 8,572
- Set.ofPredstatement · cited by 6,101
- IsClosedstatement and proof · cited by 1,639
- ClosedIicTopologystatement and proof · cited by 115
- UpperSemicontinuousOnstatement · cited by 57
- lowerSemicontinuousOn_iff_isClosed_epigraphproof · cited by 2
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