Theorems · Theorem · general topology

StrictMono.induced_topology_eq_preorder

∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst_1 : LinearOrder β] [t : TopologicalSpace β]
  [OrderTopology β] {f : α → β},
  StrictMono f → (Set.range f).OrdConnected → TopologicalSpace.induced f t = Preorder.topology α

The topology induced by a strictly monotone function with order-connected range is the preorder topology.

Defined in

Mathlib.Topology.Order.Basic

Cited by

1 results in Mathlib

Foundations

Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderLinearOrderTopologicalSpaceOrderTopology