Theorems · Theorem · general topology

StrictMono.isEmbedding_of_ordConnected

∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst_1 : LinearOrder β] [inst_2 : TopologicalSpace α]
  [h : OrderTopology α] [inst_3 : TopologicalSpace β] [OrderTopology β] {f : α → β},
  StrictMono f → (Set.range f).OrdConnected → Topology.IsEmbedding f

A strictly monotone function between linear orders with order topology is a topological embedding provided that the range of f is order-connected.

Defined in

Mathlib.Topology.Order.Basic

Cited by

9 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderLinearOrderTopologicalSpaceOrderTopologyTopologicalSpaceOrderTopology