Theorems · Theorem · Lie groups

AddMonoidHom.isOpenQuotientMap_of_isQuotientMap

∀ {A : Type u_1} [inst : AddGroup A] [inst_1 : TopologicalSpace A] [ContinuousAdd A] {B : Type u_2}
  [inst_3 : AddGroup B] [inst_4 : TopologicalSpace B] {F : Type u_3} [inst_5 : FunLike F A B] [AddMonoidHomClass F A B]
  {φ : F}, Topology.IsQuotientMap ⇑φ → IsOpenQuotientMap ⇑φ

Let A and B be topological additive groups, and let φ : A → B be a continuous surjective additive group homomorphism. Assume furthermore that φ is a quotient map (i.e., V ⊆ B is open iff φ⁻¹ V is open). Then φ is an open quotient map, and in particular an open map.

Defined in

Mathlib.Topology.Algebra.Group.Basic

Cited by

3 results in Mathlib

Foundations

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

Assumes

AddGroupTopologicalSpaceContinuousAddAddGroupTopologicalSpaceFunLikeAddMonoidHomClass