map_topologicalClosure_le — Mathlib

English

If two subalgebras s and s1 are equal, then their topological closures are equal.

Русский

Если две подалгебры s и s1 равны, то их топологические замыкания равны.

LaTeX

$$Eq s s1 → Eq (s.topologicalClosure) (s1.topologicalClosure)$$

Lean4

/-- Under a continuous algebra map, the image of the `TopologicalClosure` of a subalgebra is
contained in the `TopologicalClosure` of its image. -/
theorem _root_.Subalgebra.map_topologicalClosure_le [IsTopologicalSemiring B] (f : A →A[R] B) (s : Subalgebra R A) :
    map f s.topologicalClosure ≤ (map f.toAlgHom s).topologicalClosure :=
  image_closure_subset_closure_image f.continuous

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