{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261018-r01-topological-space-0039",
  "pilot_number": "R0017-09289",
  "number": 9289,
  "family_id": "qf-ext18-topology-open-closed-inventory",
  "title": {
    "zh-Hant": "第 9289 題：有限拓撲完整核對",
    "en": "Question 9289: Every Open, Closed, and Clopen Set in a Finite Topology"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261018-r01-topological-space-0039",
    "batch_id": "batch-20261013-advance-20261018-r01-topological-space-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-topological-space"
    ],
    "title_zh": "有限拓撲完整核對",
    "stem_zh": "有限底集合完整資料。操作=topology-open-closed-inventory；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 的欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])",
      "explanation_zh": "擴展基礎核對=(4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；抽查不構成拓撲公理；開集族與底集合分離。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有限拓撲完整核對"
      ]
    },
    "variant": {
      "family_id": "qf-ext18-topology-open-closed-inventory",
      "seed": 202610181039,
      "parameters": {
        "n": 4,
        "opens": [
          [],
          [
            1
          ],
          [
            0,
            1
          ],
          [
            0,
            1,
            3
          ],
          [
            0,
            1,
            2,
            3
          ]
        ],
        "domain": "Fin_topology_family",
        "operation": "topology-open-closed-inventory",
        "semantic_template": "qf-ext18-topology-open-closed-inventory/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261018-r01-topological-space-0039",
    "batch_id": "batch-20261013-advance-20261018-r01-topological-space-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "24532b1147167975477b066e1ccb75021a9bef234ed8917253201aaca2cf6dfc",
      "family_id": "qf-ext18-topology-open-closed-inventory",
      "seed": 202610181039,
      "semantic_template": "qf-ext18-topology-open-closed-inventory/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]]}"
    },
    "fields": {
      "title": "Every Open, Closed, and Clopen Set in a Finite Topology",
      "stem": "Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.",
      "answer": {
        "kind": "expression",
        "value": "(4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])",
        "explanation": "Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."
      },
      "skills": [
        "Every Open, Closed, and Clopen Set in a Finite Topology"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "有限底集合完整資料。操作=topology-open-closed-inventory；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 的欄序回答。",
        "en": "Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "space=given;n=4;raw family=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]",
      "after": "universe=[0, 1, 2, 3];canonical opens=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]",
      "relation": "implies",
      "rule_id": "finite_topology_normalize_relative_universe",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個候選先合型為同一Fin(n)的子集，去重元素及重複子集，再以二進位mask排序；開集族與底集合是不同層資料。",
        "en": "Type every candidate as a subset of the same Fin(n),deduplicate elements and members,and order by binary masks.The open family and underlying set are distinct data layers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "space=given;empty=是;full=是;union closed=是;intersection closed=是",
      "after": "topology=是;first union failure=無;first intersection failure=無",
      "relation": "implies",
      "rule_id": "finite_topology_axioms_joint_gate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "四條件同時成立才是拓撲。底集合有限，所以不同開集也只有有限多個；任意聯集可約成有限不同成員的聯集，由二元閉合與空聯集得到。空交集需全集，無成員時二元條件真也不能略過空集／全集。",
        "en": "All four conditions are required.On a finite underlying set there are finitely many distinct opens,so an arbitrary union reduces to a finite distinct-member union,covered by binary closure and the empty union.The empty intersection needs the whole set.Vacuous binary closure does not waive empty/whole-set membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "universe Fin(4);open=[]",
      "after": "relative complement=[0, 1, 2, 3];closed=是",
      "relation": "implies",
      "rule_id": "closed_as_relative_complement_of_open",
      "conditions": [],
      "explanation": {
        "zh-Hant": "閉集由同一底集合中開集的補集得到，不是用『不是開集』作定義。",
        "en": "A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "universe Fin(4);open=[1]",
      "after": "relative complement=[0, 2, 3];closed=是",
      "relation": "implies",
      "rule_id": "closed_as_relative_complement_of_open",
      "conditions": [],
      "explanation": {
        "zh-Hant": "閉集由同一底集合中開集的補集得到，不是用『不是開集』作定義。",
        "en": "A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "universe Fin(4);open=[0, 1]",
      "after": "relative complement=[2, 3];closed=是",
      "relation": "implies",
      "rule_id": "closed_as_relative_complement_of_open",
      "conditions": [],
      "explanation": {
        "zh-Hant": "閉集由同一底集合中開集的補集得到，不是用『不是開集』作定義。",
        "en": "A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "universe Fin(4);open=[0, 1, 3]",
      "after": "relative complement=[2];closed=是",
      "relation": "implies",
      "rule_id": "closed_as_relative_complement_of_open",
      "conditions": [],
      "explanation": {
        "zh-Hant": "閉集由同一底集合中開集的補集得到，不是用『不是開集』作定義。",
        "en": "A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "universe Fin(4);open=[0, 1, 2, 3]",
      "after": "relative complement=[];closed=是",
      "relation": "implies",
      "rule_id": "closed_as_relative_complement_of_open",
      "conditions": [],
      "explanation": {
        "zh-Hant": "閉集由同一底集合中開集的補集得到，不是用『不是開集』作定義。",
        "en": "A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "opens=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]];closed sets=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]",
      "after": "clopen sets=[[], [0, 1, 2, 3]]",
      "relation": "implies",
      "rule_id": "clopen_intersection_of_properties",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時在兩個族的集合才同時開閉；空集與全集都在，其他集合也可能同時開閉。",
        "en": "Exactly the members of both families are clopen.Empty and whole sets are included,and other sets can also be clopen."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      4,
      [
        [],
        [
          1
        ],
        [
          0,
          1
        ],
        [
          0,
          1,
          3
        ],
        [
          0,
          1,
          2,
          3
        ]
      ],
      [
        [],
        [
          2
        ],
        [
          2,
          3
        ],
        [
          0,
          2,
          3
        ],
        [
          0,
          1,
          2,
          3
        ]
      ],
      [
        [],
        [
          0,
          1,
          2,
          3
        ]
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261013-advance-20261018-r01-topological-space-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261018-r01-topological-space-foundations/questions.jsonl",
      "batch_sha256": "9d63d2019d578bf3d3a3d09635666b7dd08cd8ca93894403258ded79a01fe94f",
      "original_line": 39,
      "original_line_sha256": "6fda891095d2283f44fb041c22e07459aa939e6b48c974443f39bb971f56f497",
      "question_compact_sha256": "24532b1147167975477b066e1ccb75021a9bef234ed8917253201aaca2cf6dfc",
      "english_row_compact_sha256": "b1e298885d0570a6058f676b81f5caa26c79004a09c376d6bcda3d7187fb957b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261018-r01-topological-space-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261018-r01-topological-space-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-topological-space": {
        "sha256": "95f78b4f323b3139dd05068842f7209ac33b39b3d86feb5b73e0eb8849cdb43d",
        "english_sha256": "d4be991db50b46add5676c87b921ccea446b23b0f93e496d3dac77e5f07c3d7c",
        "original_path": "public/data/mko/topological-space.json",
        "english_path": "public/data/i18n/editions/mko-topological-space.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "20e59f30c0b6218e6d9614c16170d770e56a2f1e4ae5ce39d0a8a3db206afa61",
    "original_candidate_sha256": "99496784a378715d62a86539afe51a6455faa6eaba95de89a3fa32b8b2c8650b",
    "original_candidate_line_sha256": "99496784a378715d62a86539afe51a6455faa6eaba95de89a3fa32b8b2c8650b"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-006-p18",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8",
      "question_first_witness_sha256": "43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 8,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "1e0726f3283eb1038baa63fe05a02085e061c1992427ba3f00381d61b4f44201",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}