Theorems · Theorem · general topology

MonotoneOn.map_csSup_of_continuousWithinAt

∀ {α : Type u_1} {β : Type u_2} [inst : ConditionallyCompleteLinearOrder α] [inst_1 : TopologicalSpace α]
  [OrderTopology α] [inst_3 : ConditionallyCompleteLinearOrder β] [inst_4 : TopologicalSpace β] [OrderClosedTopology β]
  {f : α → β} {A : Set α},
  ContinuousWithinAt f A (sSup A) →
    MonotoneOn f A →
      A.Nonempty →
        autoParam (BddAbove A) MonotoneOn.map_csSup_of_continuousWithinAt._auto_1 → f (sSup A) = sSup (f '' A)

A monotone function continuous at the supremum of a nonempty set sends this supremum to the supremum of the image of this set.

Defined in

Mathlib.Topology.Order.Monotone

Cited by

4 results in Mathlib

Foundations

Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ConditionallyCompleteLinearOrderTopologicalSpaceOrderTopologyConditionallyCompleteLinearOrderTopologicalSpaceOrderClosedTopology