Theorems · Theorem · general topology

IsUniformInducing.completeSpace_congr

∀ {α : Type u} {β : Type v} [inst : UniformSpace α] [inst_1 : UniformSpace β] {f : α → β},
  IsUniformInducing f → Function.Surjective f → (CompleteSpace α ↔ CompleteSpace β)

If f is a surjective uniform inducing map, then its domain is a complete space iff its codomain is a complete space. See also _root_.completeSpace_congr for a version that assumes f to be an equivalence.

Defined in

Mathlib.Topology.UniformSpace.UniformEmbedding

Cited by

6 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

UniformSpaceUniformSpace