Theorems · Theorem · functional analysis

Subalgebra.map_topologicalClosure_le

∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : TopologicalSpace A]
  {B : Type u_3} [inst_3 : Semiring B] [inst_4 : TopologicalSpace B] [inst_5 : Algebra R A] [inst_6 : Algebra R B]
  [inst_7 : IsSemitopologicalSemiring A] [inst_8 : IsSemitopologicalSemiring B] (f : A →A[R] B) (s : Subalgebra R A),
  Subalgebra.map (↑f) s.topologicalClosure ≤ (Subalgebra.map (↑f) s).topologicalClosure

Under a continuous algebra map, the image of the TopologicalClosure of a subalgebra is contained in the TopologicalClosure of its image.

Defined in

Mathlib.Topology.Algebra.Algebra

Cited by

0 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringSemiringTopologicalSpaceSemiringTopologicalSpaceAlgebraAlgebraIsSemitopologicalSemiringIsSemitopologicalSemiring