English
Let f be a continuous semilinear map between topological modules, and let S be a submodule of the domain. Then the image under f of the topological closure of S is contained in the topological closure of the image of S: f(\overline{S}) \subseteq \overline{f(S)}.
Русский
Пусть f — непрерывное полубозное линейное отображение между топологическими модулями, и пусть S — подмодуль области. Тогда образ под閉ки по f от замыкания S содержится в замыкании образа S: f(\overline{S}) \subseteq \overline{f(S)}.
LaTeX
$$$ f\big(\overline{S}\big) \subseteq \overline{f(S)}. $$$
Lean4
/-- Under a continuous linear map, the image of the `TopologicalClosure` of a submodule is
contained in the `TopologicalClosure` of its image. -/
theorem _root_.Submodule.topologicalClosure_map [RingHomSurjective σ₁₂] [TopologicalSpace R₁] [TopologicalSpace R₂]
[ContinuousSMul R₁ M₁] [ContinuousAdd M₁] [ContinuousSMul R₂ M₂] [ContinuousAdd M₂] (f : M₁ →SL[σ₁₂] M₂)
(s : Submodule R₁ M₁) :
s.topologicalClosure.map (f : M₁ →ₛₗ[σ₁₂] M₂) ≤ (s.map (f : M₁ →ₛₗ[σ₁₂] M₂)).topologicalClosure :=
image_closure_subset_closure_image f.continuous
