{ "artifactId": "artifact.library.mathlib.tietze-extension-theorem.v001", "candidateOnly": false, "category": "Topology", "claimBoundary": "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.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "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", "family": "Extension theorems", "id": "library.mathlib.tietze-extension-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/tietze-extension-theorem/", "entryData": "/data/library-theorems/tietze-extension-theorem.json", "evidence": "/proofs/artifact.library.mathlib.tietze-extension-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.tietze-extension-theorem.v001.evidence.json", "source": "/sources/upstream/tietze-extension-theorem/" }, "landmark": { "completeFormalType": "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", "concepts": [ "Tietze extension theorem", "normal topological spaces", "closed embeddings", "bounded continuous functions", "norm-preserving extension", "Urysohn lemma", "geometric approximation" ], "plainLanguage": "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.", "poster": { "alt": "An ivory topology poster states the closed-embedding Tietze extension theorem, shows a continuous profile extending from a sealed inset across a larger normal space under shared norm bounds, and summarizes one-third corrections, two-thirds error decay, and a Cauchy limit.", "byteSize": 3087880, "caption": "A bounded continuous real-valued function extends along a closed embedding to the whole normal space with exact agreement and the same norm.", "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.", "derivatives": [ { "byteSize": 156938, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/tietze-extension-theorem/theorem-poster-v2-640.webp", "sha256": "sha256:a2fe115c884e6d87fdc8fbff188dd18a1517aeec71610d78dc5d29170a8f705f", "width": 640 }, { "byteSize": 366780, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/tietze-extension-theorem/theorem-poster-v2-1024.webp", "sha256": "sha256:7cebbdbdc722270de87fca980e83645e0a0314afd4ad317722f92b85e2d78a17", "width": 1024 } ], "generatedAt": "2026-07-25T17:02:00-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/tietze-extension-theorem/theorem-poster-v2.png", "sha256": "sha256:184188ec23227c6b9da74b7cd9816955e228ebf208dcd416f143a9c516148d86", "title": "Tietze Extension Theorem — bounded real-valued form", "width": 1024, "kind": "theorem_poster", "transcript": "TOPOLOGY · NORMAL SPACES\nTIETZE EXTENSION THEOREM\nBOUNDED REAL-VALUED FORM\nX topological · Y normal\ne : X → Y is a closed embedding\nf : X →ᵇ ℝ is bounded and continuous\nTHERE EXISTS g : Y →ᵇ ℝ\n‖g‖ = ‖f‖\ng ∘ e = f\nTHE CHECKED ROUTE\nONE-THIRD CORRECTIONS\nTWO-THIRDS ERROR DECAY\nCAUCHY LIMIT\nEXACT SCOPE\nNo uniqueness, canonical extension, constructive algorithm, arbitrary codomain, or unbounded-function claim.\nPROOF ATLAS · EXISTING MATHLIB THEOREM" }, "visual": { "alt": "A sealed ivory inset sits inside a larger green topographic surface; its crests, troughs, and contour field continue outward while shared gold upper and lower limits preserve the same range.", "byteSize": 3301749, "caption": "The bounded real-valued profile extends from the closed embedded copy to the whole normal space while retaining exact agreement and the same norm.", "description": "A sealed central inset distinguishes the original closed embedded domain from the larger ambient surface. The continuous topographic field opens across the surrounding space under one shared pair of gold norm limits, with faint contour echoes suggesting the geometrically shrinking corrections used by the checked source.", "derivatives": [ { "byteSize": 53568, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/tietze-extension-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:1bde587d0db11c30f1c756395f62e2eaf355fd7138b2e2be01a47d2629c94407", "width": 640 }, { "byteSize": 220302, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/tietze-extension-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:efa1c5f4a5408beeba5839099ce9a05f921b4daa217cac7bb68003faaffd1770", "width": 1200 } ], "generatedAt": "2026-07-25T20:23:08Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/tietze-extension-theorem/theorem-schematic-v1.png", "sha256": "sha256:561980182c7b8c2295179b47310f8174f22a26f5dde0526e012de65e478e3acf", "title": "A closed profile continues under the same norm bounds", "width": 1536 }, "walkthrough": null, "whyLandmark": "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." }, "nonClaims": [ "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." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.tietze-extension-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "Every bounded continuous real-valued function along a closed embedding into a normal space has a same-norm bounded continuous extension.", "targetId": "target.library.mathlib.tietze-extension-theorem.v001", "title": "Tietze Extension Theorem — Bounded Real-Valued Form", "upstreamOrigin": { "declarationName": "BoundedContinuousFunction.exists_extension_norm_eq_of_isClosedEmbedding", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Topology/TietzeExtension.lean#L272", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:1770fba1f2b2df6aaa098fa7c55b9033bfb18b689a70a6afbf8cc253606826e2", "sourceByteLength": 29183, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Topology/TietzeExtension.lean", "sourceLine": 272, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }