Theorems · Theorem · general topology

upperHemicontinuous_iff_isOpen_preimage_Iic

∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] {f : α → Set β} [inst_1 : TopologicalSpace β],
  UpperHemicontinuous f ↔ ∀ (u : Set β), IsOpen u → IsOpen (f ⁻¹' Set.Iic u)

A correspondence f : α → Set β is upper hemicontinuous if and only if its upper inverse (i.e., u : Set β ↦ f ⁻¹' (Iic u), note that f ⁻¹' (Iic u) = {x | f x ⊆ u}) sends open sets to open sets.

Defined in

Mathlib.Topology.Semicontinuity.Hemicontinuity

Cited by

2 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace