{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-compact-subset-closed-in-hausdorff",
  "semantic_family_id": "sf-compact-subset-closed-in-hausdorff",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Hausdorff空間中的緊緻集為閉集"
  },
  "summary": {
    "zh-Hant": "若K是Hausdorff空間X的緊緻子集，則K在X中為閉集。"
  },
  "statement": {
    "zh-Hant": "對每個x∉K及每個y∈K，以Hausdorff性選不交open neighborhoods U_y含x與V_y含y；{V_y}覆蓋K，compactness給有限subcover。相應有限個U_y的交集是與K不交的open neighborhood of x，故X\\K open。empty compact subset亦closed。Hausdorff假設不可刪；converse也不成立。"
  },
  "formula": {
    "tex": "C(K)=1",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>C</mi><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "C",
          "glyph": "C"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "K",
            "glyph": "K"
          }
        ]
      },
      "rhs": {
        "type": "number",
        "value": 1
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ebc4a0330370419091b83c6c089a71c14bc05c4db27151fb7c39771c83bc1cd3"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "精確題族使用完整有限carrier與完整topology，逐對核Hausdorff並逐集合核open-cover責任。"
    },
    {
      "id": "a2",
      "zh-Hant": "finite-topology certificates只證明指定有限實例；一般定理邊界仍保留ambient Hausdorff與compact子集兩項假設。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "closedness_predicate",
      "role_zh": "閉集述詞",
      "scope": "representative_formula"
    },
    {
      "token": "K",
      "role": "compact_subset",
      "role_zh": "緊緻子集",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-hausdorff-space",
      "title_zh": "Hausdorff空間",
      "reason_zh": "不同點的互斥open neighborhoods。"
    },
    {
      "id": "mko-compact-subset",
      "title_zh": "緊緻子集",
      "reason_zh": "open cover的有限subcover。"
    },
    {
      "id": "mko-open-closed-set",
      "title_zh": "開集與閉集",
      "reason_zh": "補集open等價於closed。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "點對點分離先形成覆蓋K的neighborhood family。",
      "compactness把可能無限的覆蓋縮成有限個，才能交疊x側open sets。",
      "non-Hausdorff空間可有compact但非closed子集；closed也不必compact。"
    ],
    "common_misconception_zh": "只用一對neighborhood就覆蓋整個K，或遺漏Hausdorff假設便宣稱所有compact subsets closed。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限模型核對不是一般證明替代品；逆命題及去除Hausdorff的版本均不成立。",
    "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 Hausdorff與局部緊緻性。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-compact-subset-closed-in-hausdorff.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-11-12 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"
  }
}