Theorems · Theorem · general topology
upperSemicontinuous_iff_isClosed_preimage
∀ {α : Type u_1} [inst : TopologicalSpace α] {γ : Type u_4} [inst_1 : LinearOrder γ] {f : α → γ},
UpperSemicontinuous f ↔ ∀ (y : γ), IsClosed (f ⁻¹' Set.Ici y)- Defined in
- Mathlib.Topology.Semicontinuity.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceLinearOrder
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.
- TopologicalSpacestatement and proof · cited by 24,529
- LinearOrderstatement and proof · cited by 8,572
- Set.preimagestatement and proof · cited by 4,946
- IsOpenproof · cited by 2,400
- IsClosedstatement and proof · cited by 1,639
- Set.Iioproof · cited by 1,166
- Set.Icistatement and proof · cited by 1,070
- UpperSemicontinuousstatement · cited by 52
- Set.compl_Iciproof · cited by 11
- upperSemicontinuous_iff_isOpen_preimageproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- UpperSemicontinuous.isClosed_preimageproof · cited by 0