Theorems · Theorem · general topology

ContinuousMap.exists_restrict_eq_forall_mem_of_closed

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

Tietze extension theorem for real-valued continuous maps, a version for a closed set. Let s be a closed set in a normal topological space Y. Let f be a 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 continuous real-valued function g : C(Y, ℝ) such that g y ∈ t for all y and g.restrict s = f.

Defined in

Mathlib.Topology.TietzeExtension

Cited by

0 results in Mathlib

Foundations

Depth 185 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceNormalSpaceSet.OrdConnected