Theorems · Theorem · general topology

AffineMap.isOpenMap_linear_iff

∀ {R : Type u_1} {V : Type u_2} {P : Type u_3} {W : Type u_4} {Q : Type u_5} [inst : AddCommGroup V]
  [inst_1 : TopologicalSpace V] [inst_2 : AddTorsor V P] [inst_3 : TopologicalSpace P] [IsTopologicalAddTorsor P]
  [inst_5 : AddCommGroup W] [inst_6 : TopologicalSpace W] [inst_7 : AddTorsor W Q] [inst_8 : TopologicalSpace Q]
  [IsTopologicalAddTorsor Q] [inst_10 : Ring R] [inst_11 : Module R V] [inst_12 : Module R W] {f : P →ᵃ[R] Q},
  IsOpenMap ⇑f.linear ↔ IsOpenMap ⇑f

If f is an affine map, then its linear part is an open map iff f is an open map.

Defined in

Mathlib.Topology.Algebra.Affine

Cited by

2 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupTopologicalSpaceAddTorsorTopologicalSpaceIsTopologicalAddTorsorAddCommGroupTopologicalSpaceAddTorsorTopologicalSpaceIsTopologicalAddTorsorRingModuleModule