Theorems · Theorem · general topology

DiscreteTopology.of_continuous_injective

∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [DiscreteTopology β]
  {f : α → β}, Continuous f → Function.Injective f → DiscreteTopology α

If the codomain of a continuous injective function has discrete topology, then so does the domain. See also Embedding.discreteTopology for an important special case.

Defined in

Mathlib.Topology.Order

Cited by

2 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceDiscreteTopology