Existing in Mathlib · Topology

Tietze Extension Theorem — Bounded Real-Valued Form

A bounded continuous real-valued function on a closed embedded copy of a space can be continued across the whole normal ambient space without changing its values there or its norm.

Tietze extension theorem
normal topological spaces
closed embeddings
bounded continuous functions
norm-preserving extension
Urysohn lemma
geometric approximation

Exact theorem

Exact Mathlib statement

theorem BoundedContinuousFunction.exists_extension_norm_eq_of_isClosedEmbedding {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [NormalSpace Y] (f : X →ᵇ ℝ) {e : X → Y} (he : IsClosedEmbedding e) : ∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g ∘ e = f

The theorem at a glance

Tietze Extension Theorem — bounded real-valued form

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

A bounded continuous real-valued function extends along a closed embedding to the whole normal space with exact agreement and the same norm. Explanatory diagram.

Detailed visual description

The theorem block retains the exact composition and norm-equality conclusion. A sealed embedded region shares one profile and one gold norm envelope with the surrounding normal space, while the source-visible proof route is summarized by one-third corrections, two-thirds error decay, and a Cauchy limit.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

The extension agrees with f on the closed embedded copy and has the same norm; the inset illustrates the proof's geometric error decay, not a unique or canonical extension. Explanatory scientific diagram.

Detailed visual description

On the left, a black outline labelled X encloses a blue bounded profile f. An arrow labelled e leads to the corresponding red-edged closed inset e(X) within the larger green outline marked Y normal; a blue profile g extends across Y and agrees with the inset profile. Gold horizontal guides and the real-value axis indicate the common norm bounds. A box below states g ∘ e = f and ‖g‖ = ‖f‖. At lower right, nested blue rings are labelled 1, 2/3, (2/3)², and 0, illustrating geometric error bounds tending to zero in the proof's successive approximations. These schematic surfaces and rings do not assert a metric or Euclidean structure on the general spaces. The theorem is for bounded continuous real-valued functions and a closed embedding into a normal space; it asserts neither uniqueness nor a canonical or constructive extension algorithm.

Why it matters

A mathematical landmark

Tietze extension is a central theorem of general topology: normality lets real-valued continuous data on a closed part continue across the ambient space. Mathlib's selected bounded form makes the quantitative strength explicit by preserving the supremum norm exactly.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's bounded real-valued Tietze extension theorem in closed-embedding form. For topological spaces X and Y with Y normal, a bounded continuous f : X →ᵇ ℝ and a closed embedding e : X → Y admit a bounded continuous g : Y →ᵇ ℝ with ‖g‖ = ‖f‖ and g ∘ e = f. It is not the unbounded-function theorem, an arbitrary-codomain theorem, a uniqueness statement, or a canonical or constructive extension algorithm.

The selected declaration concerns bounded continuous functions with codomain ℝ.
Its domain enters the normal ambient space through a closed embedding.
It proves existence with exact norm equality and exact bundled composition, not uniqueness or canonicity.
It does not provide a constructive numerical extension algorithm.
It is not the file's separate unbounded-function or arbitrary TietzeExtension-codomain formulation.
ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics.

Source and local evidence

Where the theorem comes from

Existing declaration: BoundedContinuousFunction.exists_extension_norm_eq_of_isClosedEmbedding in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.tietze-extension-theorem.v001
Source: Open the pinned upstream reference