Theorems · Theorem · general topology

MonotoneOn.map_sSup_of_continuousWithinAt

∀ {α : Type u_1} {β : Type u_2} [inst : CompleteLinearOrder α] [inst_1 : TopologicalSpace α] [OrderTopology α]
  [inst_3 : CompleteLinearOrder β] [inst_4 : TopologicalSpace β] [OrderClosedTopology β] {f : α → β} {s : Set α},
  ContinuousWithinAt f s (sSup s) → MonotoneOn f s → f ⊥ = ⊥ → f (sSup s) = sSup (f '' s)

A monotone function f sending bot to bot and continuous at the supremum of a set sends this supremum to the supremum of the image of this set.

Defined in

Mathlib.Topology.Order.Monotone

Cited by

3 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CompleteLinearOrderTopologicalSpaceOrderTopologyCompleteLinearOrderTopologicalSpaceOrderClosedTopology