{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-compact-subset-01",
  "mko_id": "mko-compact-subset",
  "title": {
    "zh-Hant": "有限子空間的緊緻性",
    "en": "Compactness of a finite subspace"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-compact-subset-01",
    "source": {
      "mko_id": "mko-compact-subset",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "b4f23af78bd1197e0c8a2e10786c521b3e087c5ec8033bdec3c84135b9f380fd",
      "english_sha256": "d9367471b34cd6cfb707454aaba2ecd52662b39d867d6a3407ccd260560d1856"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "sets",
      "topology",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "證 K={b,c} 在指定母空間中緊緻，並核相對開覆蓋。",
      "en": "Prove K={b,c} compact in its ambient space and audit a relative-open cover."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X={a,b,c}, tau={空集,{a},{a,b},X}, K={b,c}。",
          "en": "X={a,b,c}, tau={empty,{a},{a,b},X}, K={b,c}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "\\varnothing\\subset\\{a\\}\\subset\\{a,b\\}\\subset X",
        "after_tex": "\\tau\\text{ is a topology}",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "含兩端的有限鏈對聯交封閉。",
          "en": "The finite chain with both endpoints is closed under unions and intersections."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "核的是完整四項族。",
            "en": "The full four-member family is checked."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "U\\cap K\\ (U\\in\\tau)",
        "after_tex": "\\varnothing,\\varnothing,\\{b\\},K",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐個母開集相交，保留全部四項來源。",
          "en": "Intersect each ambient open, retaining all four source entries."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "相對開集見證不需包含於 K。",
            "en": "Ambient witnesses need not be subsets of K."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S03",
        "uses": [
          "S02"
        ],
        "before_tex": "\\tau_K=\\{\\varnothing,\\{b\\},K\\}",
        "after_tex": "\\tau_K\\text{ is a topology on }K",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "去重後仍是含空集與 K 的有限鏈。",
          "en": "After deduplication this remains a finite chain containing empty and K."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02"
          ],
          "explanation": {
            "zh-Hant": "全集在此步已改為 K。",
            "en": "The universe at this step is K."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S04",
        "uses": [
          "S03"
        ],
        "before_tex": "K\\subseteq\\bigcup_{i\\in I}V_i,\\quad V_i\\in\\tau_K",
        "after_tex": "c\\in V_j\\text{ for some }j\\in I",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任取相對開覆蓋，c 必落在某一原成員中。",
          "en": "For an arbitrary relative-open cover, c belongs to an original member."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03"
          ],
          "explanation": {
            "zh-Hant": "j 依覆蓋選取，覆蓋並未固定為單一測試。",
            "en": "j depends on the cover; the cover is not a single fixed test."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S05",
        "uses": [
          "S03",
          "S04"
        ],
        "before_tex": "c\\in V_j\\in\\tau_K",
        "after_tex": "V_j=K",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整相對族只有 K 含 c。",
          "en": "Only K in the complete relative family contains c."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "窮盡三個相對開集才可此推論。",
            "en": "The inference uses all three relative opens."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S06",
        "uses": [
          "S04",
          "S05"
        ],
        "before_tex": "J=\\{j\\}\\subseteq I,\\quad\\bigcup_{i\\in J}V_i=K",
        "after_tex": "K\\text{ is compact in the subspace topology}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個相對開覆蓋都有一項原成員子覆蓋，完成全稱定義。",
          "en": "Every relative-open cover has a one-member original subcover, establishing the universal definition."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "不依 K 是否母空間開或閉。",
            "en": "This does not depend on whether K is ambient open or closed."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S03"
        ],
        "before_tex": "(V_1,V_2)=(\\{b\\},K)",
        "after_tex": "J=\\{2\\}\\text{ covers }K;\\quad\\{1\\}\\text{ does not}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "指定覆蓋也可回讀原索引；單取 {b} 漏 c。",
          "en": "The supplied cover admits index {2}; choosing only {b} misses c."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "此特例與全稱證明分開。",
            "en": "This instance is separate from the universal proof."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S08",
        "uses": [
          "S03"
        ],
        "before_tex": "K=\\varnothing\\text{ as a separate boundary}",
        "after_tex": "\\varnothing\\text{ has the empty finite subcover}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "空子集同樣緊緻，不需從空覆蓋中捏造一個成員。",
          "en": "The empty subset is compact using the empty finite subcover, without inventing a member."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03"
          ],
          "explanation": {
            "zh-Hant": "空目標的聯集包含責任真，與非空例子不同。",
            "en": "Coverage of the empty target is vacuous, unlike the nonempty example."
          }
        },
        "leaf_basis": "sets"
      }
    ],
    "result": {
      "zh-Hant": "K 在誘導拓撲緊緻；相對開集為空集、{b}、K。",
      "en": "K is compact in the induced topology, whose opens are empty, {b} and K."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "{b} 單獨覆蓋 K。",
          "en": "{b} alone covers K."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "c 未被覆蓋。",
          "en": "c is missing."
        }
      },
      {
        "claim": {
          "zh-Hant": "任何相對開集也在母空間開。",
          "en": "Every relative open is ambient open."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "{b} 不屬母拓撲。",
          "en": "{b} is not in the ambient topology."
        }
      },
      {
        "claim": {
          "zh-Hant": "所有拓撲的緊緻等於閉且有界。",
          "en": "Compact always means closed and bounded in any topology."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本例未指定度量；一般結論需額外假設。",
          "en": "No metric is supplied, and the general equivalence needs additional premises."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "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。",
          "en": "A cover must cover K, and a subcover must be selected from the original cover. Passing one cover audit does not prove general compactness. For the complete finite topology and finite subset used here, finiteness supplies compactness, including the empty subset by convention. Do not extrapolate this finite enumeration to infinite spaces or identify compactness with ambient closedness and boundedness without extra hypotheses."
        },
        "source": {
          "mko_id": "mko-compact-subset",
          "sha256": "b4f23af78bd1197e0c8a2e10786c521b3e087c5ec8033bdec3c84135b9f380fd",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "sets",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "集合的聯、交、補集、包含與函數逆像以逐元素定義計算。",
          "en": "Compute unions, intersections, complements, inclusion and preimages element by element."
        }
      },
      {
        "id": "topology",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "拓撲含空集與全集，對任意聯及有限交封閉；有限集合族可完整核對。",
          "en": "A topology contains empty and whole sets and is closed under arbitrary unions and finite intersections; a finite family admits exhaustive checks."
        }
      },
      {
        "id": "definition_here",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "K 的相對開集為 U∩K（U 母開）；K 緊緻指每個相對開覆蓋有取自原索引的有限子覆蓋。",
          "en": "Relative opens of K are U intersect K for ambient-open U; K is compact when every relative-open cover has a finite subcover from its original indices."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整有限模型；不把緊緻普遍改寫成閉且有界。",
      "en": "A complete finite model, without a universal closed-and-bounded substitution."
    },
    "md_symbols": [
      [
        "\\tau_K",
        "誘導拓撲",
        "induced topology"
      ],
      [
        "V_i",
        "相對開覆蓋成員",
        "relative-open cover member"
      ]
    ],
    "md_error_controls": [
      {
        "title": "覆蓋在 K / Cover K",
        "zh": "相對聯集要含 K，不需含 X。",
        "en": "The relative union must contain K, not X."
      },
      {
        "title": "原索引 / Original indices",
        "zh": "有限子覆蓋保留原族成員。",
        "en": "A finite subcover retains original-family members."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "56c43748460b659d5e4882ad8b91e59c79a448b72613dbffe32f07c5de2c09e5",
    "original_source_sha256": "56c43748460b659d5e4882ad8b91e59c79a448b72613dbffe32f07c5de2c09e5",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}