{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-compact-subset",
  "semantic_family_id": "sf-compact-subset",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "緊緻子集"
  },
  "summary": {
    "zh-Hant": "K⊆X在subspace topology下compact，等價地每個relative-open cover有有限subcover。"
  },
  "statement": {
    "zh-Hant": "cover須覆蓋K，subcover只能從original cover選取。單一cover通過不證一般compactness；完整有限topology/finite subset由有限性定理compact，空集採compact convention，但不得外推infinite spaces或把compact普遍當closed-and-bounded。"
  },
  "formula": {
    "tex": "K\\subseteq X",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>K</mi><mo>⊆</mo><mi>X</mi></mrow></math>",
    "semantic_ast": {
      "type": "set_subset",
      "subset": {
        "type": "symbol",
        "name": "K",
        "glyph": "K"
      },
      "superset": {
        "type": "symbol",
        "name": "X",
        "glyph": "X"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7aaecded5d13d3118e2c85e36491fd06b8238a1d6d08f77558a056ea86b4d803"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限母拓撲、subset K與induced topology。"
    },
    {
      "id": "a2",
      "zh-Hant": "covers由完整relative-open sets構成。"
    }
  ],
  "symbols": [
    {
      "token": "K",
      "role": "compact_subset",
      "role_zh": "緊緻子集",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "ambient_topological_space",
      "role_zh": "母空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-compact-topological-space",
      "title_zh": "緊緻拓撲空間",
      "reason_zh": "open-cover definition。"
    },
    {
      "id": "mko-subspace-topology",
      "title_zh": "子空間拓撲",
      "reason_zh": "relative topology。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "compactness相對K的subspace topology。",
      "finite subcover選自原cover。",
      "finite-space theorem不外推infinite topology。"
    ],
    "common_misconception_zh": "只檢一個cover就稱compact，或普遍當closed-and-bounded。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪general positive certificate限complete finite spaces。",
    "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continuous image與Hausdorff closedness。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-compact-subset.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-11-09 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"
  }
}