exists_norm_eq_restrict_eq_of_closed

English

Define an order-preserving equivalence between opens and ideals of compact opens.

Русский

Задать взаимно однозначное соответствие между открытыми множествами и идеалами компактных открытых.

LaTeX

$$$\\text{opensEquiv}:\\mathrm{Opens}(X) \\simeq_o \\mathrm{Ideal}(\\mathrm{CompactOpens}(X))$$$

Lean4

/-- **Tietze extension theorem** for real-valued bounded continuous maps, a version for a closed
set. If `f` is a bounded continuous real-valued function defined on a closed set in a normal
topological space, then it can be extended to a bounded continuous function of the same norm defined
on the whole space. -/
theorem exists_norm_eq_restrict_eq_of_closed {s : Set Y} (f : s →ᵇ ℝ) (hs : IsClosed s) :
    ∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g.restrict s = f :=
  exists_extension_norm_eq_of_isClosedEmbedding' f ((ContinuousMap.id _).restrict s) hs.isClosedEmbedding_subtypeVal

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