Theorems · Theorem · general topology

MonotoneOn.countable_setOfPred_two_preimages

∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst_1 : TopologicalSpace α] [OrderTopology α]
  [inst_3 : LinearOrder β] {s : Set α} {f : α → β} [SecondCountableTopology α],
  MonotoneOn f s → {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}.Countable

If a function is monotone on a set in a second countable topological space, then there are only countably many points that have several preimages.

Defined in

Mathlib.Topology.Order.Monotone

Cited by

5 results in Mathlib

Foundations

Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderTopologicalSpaceOrderTopologyLinearOrderSecondCountableTopology