Theorems · Theorem · general topology
UpperSemicontinuous.isOpen_preimage
∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] {f : α → β} [inst_1 : Preorder β],
UpperSemicontinuous f → ∀ (y : β), IsOpen (f ⁻¹' Set.Iio y)- Defined in
- Mathlib.Topology.Semicontinuity.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpacePreorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Set.preimagestatement · cited by 4,946
- IsOpenstatement · cited by 2,400
- Set.Iiostatement · cited by 1,166
- UpperSemicontinuousstatement and proof · cited by 52
- upperSemicontinuous_iff_isOpen_preimageproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- UpperSemicontinuous.measurableproof · cited by 2
- exists_countable_upperSemicontinuous_isGLBproof · cited by 1