Theorems · Theorem · general topology

BoundedContinuousFunction.exists_extension_forall_mem_of_isClosedEmbedding

∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [NormalSpace Y]
  (f : BoundedContinuousFunction X ℝ) {t : Set ℝ} {e : X → Y} [hs : t.OrdConnected],
  (∀ (x : X), f x ∈ t) → t.Nonempty → Topology.IsClosedEmbedding e → ∃ g, (∀ (y : Y), g y ∈ t) ∧ ⇑g ∘ e = ⇑f

Tietze extension theorem for real-valued bounded continuous maps, a version for a closed embedding. Let e be a closed embedding of a nonempty topological space X into a normal topological space Y. Let f be a bounded continuous real-valued function on X. 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. Then there exists a bounded continuous real-valued function g : Y →ᵇ ℝ such that g y ∈ t for all y and g ∘ e = f.

Defined in

Mathlib.Topology.TietzeExtension

Cited by

2 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceNormalSpaceSet.OrdConnected