{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-homeomorphism",
  "semantic_family_id": "sf-homeomorphism",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "同胚"
  },
  "summary": {
    "zh-Hant": "同胚是雙射、連續且逆映射亦連續的映射；一般continuous bijection仍可能不是同胚。"
  },
  "statement": {
    "zh-Hant": "有限拓撲題完整列出來源與目標拓撲，逐一核目標開集的逆像及反向責任。有限離散空間任意雙射為同胚只限該範圍；同胚不升格成等距、可微或幾何全等。"
  },
  "formula": {
    "tex": "f:X\\to Y",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow></math>",
    "semantic_ast": {
      "type": "mapping",
      "function": {
        "type": "symbol",
        "name": "f",
        "glyph": "f"
      },
      "domain": {
        "type": "symbol",
        "name": "X",
        "glyph": "X"
      },
      "codomain": {
        "type": "symbol",
        "name": "Y",
        "glyph": "Y"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "28c28a16d0b842cb3d3c986cead38d14c1fbddc06efeb3c29e6e3479896c8a21"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "有限carrier、完整open-set families與total bijection全部明示。"
    },
    {
      "id": "a2",
      "zh-Hant": "continuity與inverse continuity各自使用全部開集逆像核對。"
    }
  ],
  "symbols": [
    {
      "token": "f",
      "role": "homeomorphism_candidate",
      "role_zh": "同胚候選",
      "scope": "representative_formula"
    },
    {
      "token": "f^{-1}",
      "role": "inverse_map",
      "role_zh": "逆映射",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-topological-continuity",
      "title_zh": "拓撲連續映射",
      "reason_zh": "正反向連續性。"
    },
    {
      "id": "mko-bijective",
      "title_zh": "雙射與唯一原像",
      "reason_zh": "可逆性。"
    },
    {
      "id": "mko-inverse-function",
      "title_zh": "雙射的反函數",
      "reason_zh": "反向映射。"
    },
    {
      "id": "mko-topological-space",
      "title_zh": "拓撲空間",
      "reason_zh": "完整來源與目標拓撲。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "bijection、forward continuity與inverse continuity缺一不可。",
      "完整有限拓撲可用全部開集做exact audit。",
      "離散有限結論不外推，也不保距離或微分結構。"
    ],
    "common_misconception_zh": "把任意連續雙射當同胚，或把同胚當等距與幾何全等。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪exact證書限完整有限拓撲；一般空間需獨立定理與額外假設。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入compact-to-Hausdorff continuous bijection theorem。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-homeomorphism.md 整理；2026-10-14 脈衝 R05（使用者手動接續 2026-11-05 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}