Theorems · Theorem · general topology

ContinuousOn.image_Icc

∀ {α : Type u_2} {β : Type u_3} [inst : ConditionallyCompleteLinearOrder α] [inst_1 : TopologicalSpace α]
  [OrderTopology α] [inst_3 : TopologicalSpace β] [DenselyOrdered α] [inst_5 : ConditionallyCompleteLinearOrder β]
  [OrderTopology β] {f : α → β} {a b : α},
  a ≤ b → ContinuousOn f (Set.Icc a b) → f '' Set.Icc a b = Set.Icc (sInf (f '' Set.Icc a b)) (sSup (f '' Set.Icc a b))

Defined in

Mathlib.Topology.Order.Compact

Cited by

3 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ConditionallyCompleteLinearOrderTopologicalSpaceOrderTopologyTopologicalSpaceDenselyOrderedConditionallyCompleteLinearOrderOrderTopology