{
  "artifactId": "artifact.library.mathlib.holomorphic-open-mapping-theorem.v001",
  "candidateOnly": false,
  "category": "Complex analysis",
  "claimBoundary": "This page indexes Mathlib's global holomorphic open-mapping declaration for a map g from a complex normed space E to ℂ that is analytic on a neighborhood of every point of a preconnected set U. It proves a disjunction: either g is constant on U, or the image of every ambient-open subset s contained in U is open in ℂ. It does not require U to be open, assert that g(U) is open when U is not open, state a global IsOpenMap result, or allow an arbitrary vector-valued codomain.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem AnalyticOnNhd.is_constant_or_isOpen {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] {U : Set E} {g : E → ℂ} (hg : AnalyticOnNhd ℂ g U) (hU : IsPreconnected U) : (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen s → IsOpen (g '' s)",
  "family": "Holomorphic mappings",
  "id": "library.mathlib.holomorphic-open-mapping-theorem.v001",
  "links": {
    "collection": "/collections/landmark-theorems-in-mathlib/",
    "collectionData": "/data/collections/mathlib-landmarks.json",
    "entry": "/library-theorems/holomorphic-open-mapping-theorem/",
    "entryData": "/data/library-theorems/holomorphic-open-mapping-theorem.json",
    "evidence": "/proofs/artifact.library.mathlib.holomorphic-open-mapping-theorem.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.holomorphic-open-mapping-theorem.v001.evidence.json",
    "source": "/sources/upstream/holomorphic-open-mapping-theorem/"
  },
  "landmark": {
    "completeFormalType": "theorem AnalyticOnNhd.is_constant_or_isOpen {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] {U : Set E} {g : E → ℂ} (hg : AnalyticOnNhd ℂ g U) (hU : IsPreconnected U) : (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen s → IsOpen (g '' s)",
    "concepts": [
      "holomorphic functions",
      "open mappings",
      "preconnected sets",
      "isolated zeros",
      "maximum principle",
      "complex-line restrictions"
    ],
    "plainLanguage": "An analytic map from a complex normed space into the complex plane has only two behaviors on a preconnected set: it is constant there, or it sends every ambient-open piece lying inside that set to an open region.",
    "poster": {
      "alt": "An ivory complex-analysis poster presents the exact constant-or-open alternative for an analytic map on a preconnected set, then traces isolated-zero separation, the maximum-principle target ball, a nonconstant complex-line direction, and propagation of local constancy.",
      "byteSize": 2342640,
      "caption": "Analyticity on a preconnected set forces constancy there or openness on every ambient-open piece contained in it.",
      "description": "The poster keeps the higher-dimensional complex normed domain and scalar complex codomain visible, separates the constant and open alternatives, depicts one ambient-open subset inside U, and closes with the exact warning that U itself need not be open.",
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          "width": 640
        },
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          "height": 1536,
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          "publicPath": "/assets/library-theorems/holomorphic-open-mapping-theorem/theorem-poster-v1-1024.webp",
          "sha256": "sha256:87096ae0140cf7c4555c60ace2a13d55ce32a9a311e0386c6cda278b9366e4d7",
          "width": 1024
        }
      ],
      "generatedAt": "2026-07-25T20:32:43Z",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/holomorphic-open-mapping-theorem/theorem-poster-v1.png",
      "sha256": "sha256:14da489ba0ae2e8860f35955b247930b9a842e75d086f3fefa7785b506fe45c8",
      "title": "Holomorphic Open Mapping Theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "COMPLEX ANALYSIS · HOLOMORPHIC MAPS\nHOLOMORPHIC OPEN MAPPING THEOREM\n\ng maps E into the complex plane\ng analytic near every point of a preconnected set U\nE a complex normed space\n\nEITHER\ng is constant on U\nOR\nfor every ambient-open s ⊆ U, g(s) is open in the complex plane\n\nNONCONSTANT ANALYTIC MAPS OPEN THE AMBIENT-OPEN PIECES OF U\n\nHOW THE SOURCE ROUTE MOVES\n1 · Isolated zeros give a positive boundary minimum on a complex line\n2 · The maximum principle puts a target ball inside the local image\n3 · A nonconstant line direction gives local openness in E\n4 · Preconnectedness propagates local constancy across U\n\nEXACT SCOPE\nU need not be open. No claim that g(U) is open unless U is open."
    },
    "visual": {
      "alt": "A text-free preconnected source body contains two full-dimensional open pieces whose analytic flow reaches a gold fork: one separated branch converges to a single point, while the other branches fill open regions in scalar target planes.",
      "byteSize": 2466517,
      "caption": "The analytic alternative is rigidity to one value or full-dimensional openness for every ambient-open piece inside U.",
      "description": "An abstract complex normed-space field contains one coherent preconnected region and two interior neighborhoods. A central gold fork keeps a quiet constant-point outcome separate from a dominant open-image outcome, while a complex-line slice, boundary sphere, and contained target ball remain secondary proof-route motifs.",
      "derivatives": [
        {
          "byteSize": 35306,
          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/holomorphic-open-mapping-theorem/theorem-schematic-v1-640.webp",
          "sha256": "sha256:c6bc9492d699894e67e38bb2f5e640323f964d5a574e0a060159d22e4b5a3b40",
          "width": 640
        },
        {
          "byteSize": 116890,
          "height": 800,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/holomorphic-open-mapping-theorem/theorem-schematic-v1-1200.webp",
          "sha256": "sha256:1c4ea27d11d1e7e0ffa68d0ff9f9336298a07fc71727ed05f438b103ff99ff47",
          "width": 1200
        }
      ],
      "generatedAt": "2026-07-25T20:32:43Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/holomorphic-open-mapping-theorem/theorem-schematic-v1.png",
      "sha256": "sha256:474ce6b3c1784119a0c3d039d9158b3e1f105732a786c2d2bd70541aa16541ce",
      "title": "Constant or open on every ambient-open piece",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "The open mapping theorem is a central rigidity principle of complex analysis: nonconstant analytic behavior cannot flatten genuine neighborhoods. Mathlib's selected formulation exposes both the local analytic mechanism and its global propagation across a preconnected set, while allowing a higher-dimensional complex normed domain."
  },
  "nonClaims": [
    "The set U is assumed preconnected but need not be open; the selected declaration therefore does not conclude that g(U) is open unless U is open.",
    "The open alternative ranges over ambient-open subsets s contained in U and is not itself a global IsOpenMap statement.",
    "The domain may be an arbitrary complex normed space, but the codomain in the selected declaration is exactly ℂ rather than an arbitrary complex normed space.",
    "The conclusion does not assert injectivity, conformality, biholomorphic inversion, or a quantitative distortion bound.",
    "Proof Atlas is indexing an existing Mathlib theorem, and the generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.holomorphic-open-mapping-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": "A map to ℂ analytic near every point of a preconnected set is constant there or maps every ambient-open subset contained in that set to an open subset of ℂ.",
  "targetId": "target.library.mathlib.holomorphic-open-mapping-theorem.v001",
  "title": "Holomorphic Open Mapping Theorem",
  "upstreamOrigin": {
    "declarationName": "AnalyticOnNhd.is_constant_or_isOpen",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/Complex/OpenMapping.lean#L166",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
    "sourceArtifactHash": "sha256:352dd28e4e85d7c15c69ad792b6db68d2bab2e18c652b461823002a0cba755e6",
    "sourceByteLength": 16107,
    "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6",
    "sourceFile": "Mathlib/Analysis/Complex/OpenMapping.lean",
    "sourceLine": 166,
    "verificationKind": "git_worktree"
  },
  "publicPresentationReview": {
    "status": "reviewed"
  }
}