Theorems · Theorem · general topology

Monotone.map_csInf_of_continuousAt

∀ {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},
  ContinuousAt f (sInf A) →
    Monotone f →
      A.Nonempty → autoParam (BddBelow A) Monotone.map_csInf_of_continuousAt._auto_1 → f (sInf A) = sInf (f '' A)

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

Defined in

Mathlib.Topology.Order.Monotone

Cited by

2 results in Mathlib

Foundations

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

Assumes

ConditionallyCompleteLinearOrderTopologicalSpaceOrderTopologyConditionallyCompleteLinearOrderTopologicalSpaceOrderClosedTopology