{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-compact-topological-space",
  "semantic_family_id": "sf-compact-topological-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "緊緻拓撲空間"
  },
  "summary": {
    "zh-Hant": "拓撲空間緊緻，若每個開覆蓋都有有限子覆蓋；每個有限拓撲空間皆緊緻。"
  },
  "statement": {
    "zh-Hant": "本輪完整驗證有限carrier/topology並使用有限性全稱定理：open family本身有限，故任何cover已有有限subcover。另核 supplied cover，但單一cover成功不證一般或無限空間compact。"
  },
  "formula": {
    "tex": "C(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "e1849a5c630feddeb7575d1f7a48d4caeb284f626f4bf8d15bc6472592554125"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限載體與完整拓撲。"
    },
    {
      "id": "a2",
      "zh-Hant": "一般compactness的量詞是每個open cover；無限例子不由有限抽樣處理。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "compact_topological_space",
      "role_zh": "緊緻空間證書",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "topological_space",
      "role_zh": "指定拓撲空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-topological-space",
      "title_zh": "拓撲空間",
      "reason_zh": "提供完整開集族。"
    },
    {
      "id": "mko-open-cover-and-subcover",
      "title_zh": "開覆蓋與子覆蓋",
      "reason_zh": "compactness量詞對象。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "每個cover存在有限subcover。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "有限拓撲的開集族有限。",
      "任何open cover本身即有限或可去重成有限subcover。",
      "單一cover witness不證無限空間compact。"
    ],
    "common_misconception_zh": "測試幾個cover就對一般空間宣稱compact。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "全稱正證只限完整有限空間；named infinite examples另需證書。",
    "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finite intersection property與metric compactness。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-compact-topological-space.md 整理；2026-10-14 脈衝 R05（使用者手動接續 2026-11-02 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"
  }
}