Public Mathlib landmark · existing upstream theorem · read-only
ProofAtlas

Topology · existing Mathlib declaration

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.

The theorem at a glance

Tietze Extension Theorem — bounded real-valued form

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

A bounded continuous real-valued function extends along a closed embedding to the whole normal space with exact agreement and the same norm. Generated explanation only; the exact HTML statement and pinned source are authoritative.
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.

Formal orientation

Statement map

Statement map for Tietze Extension Theorem — Bounded Real-Valued FormEvery bounded continuous real-valued function along a closed embedding into a normal space has a same-norm bounded continuous extension. The pinned upstream declaration is BoundedContinuousFunction.exists_extension_norm_eq_of_isClosedEmbedding. The exact checked statement is 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.Mathematical readingEvery bounded continuousreal-valued functionalong a closed embeddinginto a normal space hasa same-norm boundedcontinuous extension.Pinned declarationmathlib ·BoundedContinuousFunction.exists_extension_norm_eq_of_isClosedEmbeddingExact checked formtheoremBoundedContinuousFunction.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 = fStatement map for Tietze Extension Theorem — Bounded Real-Valued FormEvery bounded continuous real-valued function along a closed embedding into a normal space has a same-norm bounded continuous extension. The pinned upstream declaration is BoundedContinuousFunction.exists_extension_norm_eq_of_isClosedEmbedding. The exact checked statement is 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.Mathematical readingEvery bounded continuousreal-valued functionalong a closed embeddinginto a normal space hasa same-norm boundedcontinuous extension.Pinned declarationmathlib ·BoundedContinuousFunction.exists_extension_norm_eq_of_isClosedEmbeddingExact checked formtheoremBoundedContinuousFunction.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

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

The bounded real-valued profile extends from the closed embedded copy to the whole normal space while retaining exact agreement and the same norm. Scientific diagram · explanatory, not proof evidence.
Detailed visual description

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.

Complementary intuition

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.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

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
Selected evidence declaration text
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

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

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

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.

Provenance

Origin and evidence stay separate

Existing declaration
BoundedContinuousFunction.exists_extension_norm_eq_of_isClosedEmbedding in mathlib
Relationship
The checked artifact is the upstream declaration itself
Preferred checked artifact
artifact.library.mathlib.tietze-extension-theorem.v001
Source handling
Verified upstream reference; no mirrored source package