Theorems · Theorem · general topology

BoundedContinuousFunction.exists_forall_mem_domRestrict_eq_of_closed

∀ {Y : Type u_2} [inst : TopologicalSpace Y] [NormalSpace Y] {s : Set Y} (f : BoundedContinuousFunction ↑s ℝ),
  IsClosed s →
    ∀ {t : Set ℝ} [t.OrdConnected], (∀ (x : ↑s), f x ∈ t) → t.Nonempty → ∃ g, (∀ (y : Y), g y ∈ t) ∧ g.domRestrict s = f

Tietze extension theorem for real-valued bounded continuous maps, a version for a closed set. Let s be a closed set in a normal topological space Y. Let f be a bounded continuous real-valued function on s. Let t be a nonempty convex set of real numbers (we use OrdConnected instead of Convex to automatically deduce this argument by typeclass search) such that f x ∈ t for all x : s. Then there exists a bounded continuous real-valued function g : Y →ᵇ ℝ such that g y ∈ t for all y and g.domRestrict s = f.

Defined in

Mathlib.Topology.TietzeExtension

Cited by

1 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceNormalSpaceSet.OrdConnected