Theorems · Theorem · general topology

TendstoUniformlyOn.lowerHemicontinuousOn

∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] {f : α → Set β} {s : Set α} {ι : Type u_3}
  {F : ι → α → Set β} {l : Filter ι} [l.NeBot] [inst_2 : UniformSpace β],
  TendstoUniformlyOn F f l s → (∀ (n : ι), LowerHemicontinuousOn (F n) s) → LowerHemicontinuousOn f s

A net of lower hemicontinuous set-valued functions converging uniformly on s (along a filter l) in the Hausdorff uniformity has a lower hemicontinuous limit on s

Defined in

Mathlib.Topology.Semicontinuity.Hemicontinuity

Cited by

1 results in Mathlib

Foundations

Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceFilter.NeBotUniformSpace