exists_extension_norm_eq_of_isClosedEmbedding

English

A sigma-constructed prespectral property transfers via preserved topological basis.

Русский

Свойство prespectral, построенное через сигма-конструкцию, сохраняется через базис.

LaTeX

$$$\\mathrm{PrespectralSpace}(X) \\Rightarrow \\mathrm{PrespectralSpace}(\\Sigma_i X_i)$$$

Lean4

/-- **Tietze extension theorem** for real-valued bounded continuous maps, a version with a closed
embedding and unbundled composition. If `e : C(X, Y)` is a closed embedding of a topological space
into a normal topological space and `f : X →ᵇ ℝ` is a bounded continuous function, then there exists
a bounded continuous function `g : Y →ᵇ ℝ` of the same norm such that `g ∘ e = f`. -/
theorem exists_extension_norm_eq_of_isClosedEmbedding (f : X →ᵇ ℝ) {e : X → Y} (he : IsClosedEmbedding e) :
    ∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g ∘ e = f :=
  by
  rcases exists_extension_norm_eq_of_isClosedEmbedding' f ⟨e, he.continuous⟩ he with ⟨g, hg, rfl⟩
  exact ⟨g, hg, rfl⟩

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