Theorems · Theorem · general topology

le_on_closure

∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : Preorder α] [t : OrderClosedTopology α]
  [inst_2 : TopologicalSpace β] {f g : β → α} {s : Set β},
  (∀ x ∈ s, f x ≤ g x) → ContinuousOn f (closure s) → ContinuousOn g (closure s) → ∀ ⦃x : β⦄, x ∈ closure s → f x ≤ g x

Defined in

Mathlib.Topology.Order.OrderClosed

Cited by

3 results in Mathlib

Foundations

Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpacePreorderOrderClosedTopologyTopologicalSpace