{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-compact-topological-space-01",
  "mko_id": "mko-compact-topological-space",
  "title": {
    "zh-Hant": "有限拓撲的全稱緊緻證書",
    "en": "A universal compactness certificate for a finite topology"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-compact-topological-space-01",
    "source": {
      "mko_id": "mko-compact-topological-space",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "13f302a171c38a633a5a145da984410cab9bef79a09c4aef91bdaf9e016226e6",
      "english_sha256": "cb6c5eefd1b67e492a6fcd011cf21c954ed3dcfdf9a15d5bbc46db1460c8e204"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "sets",
      "topology",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "證明三點鏈狀拓撲對每個開覆蓋都有有限子覆蓋。",
      "en": "Prove every open cover of the three-point chain topology has a finite subcover."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X={a,b,c}，tau={空集,{a},{a,b},X}；覆蓋可為任意索引族 (U_i)_{i∈I}。",
          "en": "X={a,b,c}, tau={empty,{a},{a,b},X}; a cover may be any indexed family (U_i) indexed by I."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "\\varnothing\\subset\\{a\\}\\subset\\{a,b\\}\\subset X",
        "after_tex": "\\tau\\text{ is a complete topology with }4\\text{ members}",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "有限鏈含兩端，非空聯交取最大最小項，核得全部拓撲公理。",
          "en": "The finite chain includes both endpoints; nonempty unions/intersections take largest/smallest members and verify every axiom."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "完整族的有限性已核，不是有限抽樣。",
            "en": "Finiteness holds for the complete family, not a sample."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "X=\\bigcup_{i\\in I}U_i,\\quad U_i\\in\\tau",
        "after_tex": "c\\in U_j\\text{ for some }j\\in I",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對任意覆蓋，c 必被某個原成員覆蓋。",
          "en": "For an arbitrary cover, some original member must cover c."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "選出的 j 可以依覆蓋而變，符合每覆蓋存在子覆蓋的量詞。",
            "en": "The selected j may depend on the cover, matching the required quantifier order."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S03",
        "uses": [
          "S01",
          "S02"
        ],
        "before_tex": "c\\in U_j\\in\\tau",
        "after_tex": "U_j=X",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整四項清單中只有 X 含 c。",
          "en": "Only X in the complete four-member family contains c."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "這是窮盡開集後的結論，非單一成員抽查。",
            "en": "This follows from exhaustive open-set classification, not one inspected member."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S04",
        "uses": [
          "S03"
        ],
        "before_tex": "J=\\{j\\}\\subseteq I",
        "after_tex": "\\bigcup_{i\\in J}U_i=X,\\quad |J|=1",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "保留原索引 j，得到一項的有限子覆蓋。",
          "en": "Retain the original index j to obtain a one-member finite subcover."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03"
          ],
          "explanation": {
            "zh-Hant": "沒有用不在原族的集合替換 U_j。",
            "en": "U_j has not been replaced by a set absent from the original family."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S05",
        "uses": [
          "S02",
          "S04"
        ],
        "before_tex": "\\forall\\text{ open covers }(U_i)_{i\\in I},\\ \\exists J\\subseteq I\\text{ finite}",
        "after_tex": "X\\text{ is compact}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "覆蓋最初任意，故不是僅對一次覆蓋成功，而是完成緊緻定義。",
          "en": "The starting cover was arbitrary, so this establishes the definition, not merely one successful audit."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "關鍵前提是任意性，不能由一次數值成功反推。",
            "en": "Arbitrariness is the key premise and cannot be recovered from one numerical success."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S06",
        "uses": [
          "S01"
        ],
        "before_tex": "D=\\{U_i:i\\in I\\}\\subseteq\\tau",
        "after_tex": "|D|\\le4",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "另一種一般有限族證法是對不同開集去重，再各選一個原索引。",
          "en": "Another finite-family proof deduplicates distinct opens and selects one original index for each."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01"
          ],
          "explanation": {
            "zh-Hant": "任意多次重複不會增加不同開集的數目。",
            "en": "Arbitrarily repeated members do not increase the number of distinct opens."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S07",
        "uses": [
          "S01"
        ],
        "before_tex": "(\\{a\\},\\{a,b\\},X)",
        "after_tex": "\\{X\\}\\text{ is a subcover of this supplied cover}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "指定三項覆蓋回讀第三項，與全稱證書相容但不取代它。",
          "en": "The third member of this supplied cover gives a subcover, consistent with but not replacing the universal proof."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01"
          ],
          "explanation": {
            "zh-Hant": "這一步特例和 S05 的全稱結論來自不同層級。",
            "en": "This instance and the universal conclusion at S05 are different evidence levels."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S08",
        "uses": [
          "S05"
        ],
        "before_tex": "X=\\varnothing\\text{ in the empty-space boundary}",
        "after_tex": "J=\\varnothing\\text{ covers }X",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "空空間以空子覆蓋滿足同一定義，明示邊界慣例。",
          "en": "For the empty-space boundary the empty subcover satisfies the same definition."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05"
          ],
          "explanation": {
            "zh-Hant": "只處理空載體的獨立邊界，未把本例 X 換掉。",
            "en": "This is a separate empty-carrier boundary, not a replacement of this example's X."
          }
        },
        "leaf_basis": "sets"
      }
    ],
    "result": {
      "zh-Hant": "此完整有限拓撲緊緻；證明對任意索引覆蓋成立。",
      "en": "This complete finite topology is compact, with a proof for arbitrary indexed covers."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "抽查三個覆蓋足以證任意空間緊緻。",
          "en": "Testing three covers proves any space compact."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "未遍及所有覆蓋。",
          "en": "It does not quantify over all covers."
        }
      },
      {
        "claim": {
          "zh-Hant": "原覆蓋不含 X 也可任意把 X 加進子覆蓋。",
          "en": "One may add X to a subcover even if absent from the original cover."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "子覆蓋必選原成員；此例是先證原族實含 X。",
          "en": "A subcover uses original members; here X was proved present."
        }
      },
      {
        "claim": {
          "zh-Hant": "有限開集樣本代表完整無限空間拓撲。",
          "en": "A finite sample of opens represents an entire infinite-space topology."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "有限完整性是本證書前提，不可外推。",
          "en": "Completeness and finiteness of the family are premises, not extrapolations."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "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。",
          "en": "Validate the complete finite carrier and topology, then use the universal finiteness argument: there are finitely many distinct open sets, so every open cover has a finite subcover. Supplied-cover audits are separate; success on one cover does not establish compactness of an arbitrary or infinite space."
        },
        "source": {
          "mko_id": "mko-compact-topological-space",
          "sha256": "13f302a171c38a633a5a145da984410cab9bef79a09c4aef91bdaf9e016226e6",
          "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": "緊緻：對每個索引開覆蓋 (U_i)_{i∈I}，存在有限 J⊆I 使原成員 (U_j)_{j∈J} 仍覆蓋全空間。",
          "en": "Compactness: for every indexed open cover (U_i) indexed by I, a finite J contained in I selects original members still covering the whole space."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限空間的全稱論證，另核指定覆蓋；不分類無限空間。",
      "en": "A universal finite-space argument plus one cover audit; no classification of infinite spaces."
    },
    "md_symbols": [
      [
        "I",
        "任意覆蓋索引集",
        "arbitrary cover index set"
      ],
      [
        "J",
        "有限原索引子集",
        "finite subset of original indices"
      ]
    ],
    "md_error_controls": [
      {
        "title": "全稱入口 / Universal entry",
        "zh": "先固定任意覆蓋，再選依覆蓋而定的原索引。",
        "en": "Start with an arbitrary cover and choose original indices depending on it."
      },
      {
        "title": "有限與抽樣 / Finite versus sampled",
        "zh": "完整有限拓撲的證書不是有限測試推一般結論。",
        "en": "A complete finite-topology certificate is not a general conclusion from finite tests."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
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    "original_source_sha256": "bb1992c96e3fb1b5296d4e74a562981227df547020a5dd2037c3b6c406bd4dff",
    "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."
  }
}