{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261103-r05-connected-topological-space-0055",
  "pilot_number": "R0033-17355",
  "number": 17355,
  "family_id": "qf-mod34-connected-exhaustive",
  "title": {
    "zh-Hant": "第 17355 題：連通拓撲空間",
    "en": "Question 17355: Exhaust the Complete Finite Topology before Certifying Connectedness"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261103-r05-connected-topological-space-0055",
    "batch_id": "batch-20261014-advance-20261103-r05-connected-topological-space-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-connected-topological-space"
    ],
    "title_zh": "連通拓撲空間",
    "stem_zh": "完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)",
      "explanation_zh": "Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "連通拓撲空間"
      ]
    },
    "variant": {
      "family_id": "qf-mod34-connected-exhaustive",
      "seed": 202611035055,
      "parameters": {
        "n": 5,
        "topology": [
          [],
          [
            0
          ],
          [
            0,
            1
          ],
          [
            0,
            1,
            4
          ],
          [
            0,
            1,
            2,
            4
          ],
          [
            0,
            1,
            2,
            3,
            4
          ]
        ],
        "domain": "Complete_finite_topology_connectedness",
        "operation": "connected-exhaustive",
        "semantic_template": "qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261103-r05-connected-topological-space-0055",
    "batch_id": "batch-20261014-advance-20261103-r05-connected-topological-space-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1fddb1136c66d63ec774c3ade12638db26a79213f7e2dd557a4724d1aa09593e",
      "family_id": "qf-mod34-connected-exhaustive",
      "seed": 202611035055,
      "semantic_template": "qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"
    },
    "fields": {
      "title": "Exhaust the Complete Finite Topology before Certifying Connectedness",
      "stem": "Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.",
      "answer": {
        "kind": "expression",
        "value": "([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)",
        "explanation": "Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)."
      },
      "skills": [
        "Exhaust the Complete Finite Topology before Certifying Connectedness"
      ],
      "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": "完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。",
        "en": "Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "ACTUAL carrier Fin(5);original ordered complete topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]",
      "after": "unique empty/whole opens present;empty carrier allowed;first union failure=null;first intersection failure=null",
      "relation": "implies",
      "rule_id": "connected_actual_complete_topology_including_empty_carrier",
      "conditions": [],
      "explanation": {
        "zh-Hant": "n可為0；那時空集與全集是同一個開集，不要求兩個不同集合。逐原子集與全部二元聯／交核拓撲；有限族給任意聯封閉。實際空間由這份資料固定，不由reference label改寫。",
        "en": "The actual n may be0,when empty and whole are the same open rather than two distinct sets.Check canonical subsets and all binary union/intersection obligations;the finite family then handles arbitrary unions.This data fixes the actual space,not a reference label."
      },
      "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": "original open index=0;open=[];ACTUAL same-carrier complement=[0, 1, 2, 3, 4]",
      "after": "complement open=是;nonempty proper subset=否;nontrivial clopen obstruction=否",
      "relation": "implies",
      "rule_id": "connected_every_actual_open_complement_not_selected_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原集合已開，再核補集開才是clopen；且必非空非全集才給分離。按拓撲原序列全體，不能由一個失敗候選推全域connected。",
        "en": "The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness."
      },
      "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": "original open index=1;open=[0];ACTUAL same-carrier complement=[1, 2, 3, 4]",
      "after": "complement open=否;nonempty proper subset=是;nontrivial clopen obstruction=否",
      "relation": "implies",
      "rule_id": "connected_every_actual_open_complement_not_selected_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原集合已開，再核補集開才是clopen；且必非空非全集才給分離。按拓撲原序列全體，不能由一個失敗候選推全域connected。",
        "en": "The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness."
      },
      "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": "original open index=2;open=[0, 1];ACTUAL same-carrier complement=[2, 3, 4]",
      "after": "complement open=否;nonempty proper subset=是;nontrivial clopen obstruction=否",
      "relation": "implies",
      "rule_id": "connected_every_actual_open_complement_not_selected_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原集合已開，再核補集開才是clopen；且必非空非全集才給分離。按拓撲原序列全體，不能由一個失敗候選推全域connected。",
        "en": "The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness."
      },
      "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": "original open index=3;open=[0, 1, 4];ACTUAL same-carrier complement=[2, 3]",
      "after": "complement open=否;nonempty proper subset=是;nontrivial clopen obstruction=否",
      "relation": "implies",
      "rule_id": "connected_every_actual_open_complement_not_selected_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原集合已開，再核補集開才是clopen；且必非空非全集才給分離。按拓撲原序列全體，不能由一個失敗候選推全域connected。",
        "en": "The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness."
      },
      "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": "original open index=4;open=[0, 1, 2, 4];ACTUAL same-carrier complement=[3]",
      "after": "complement open=否;nonempty proper subset=是;nontrivial clopen obstruction=否",
      "relation": "implies",
      "rule_id": "connected_every_actual_open_complement_not_selected_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原集合已開，再核補集開才是clopen；且必非空非全集才給分離。按拓撲原序列全體，不能由一個失敗候選推全域connected。",
        "en": "The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness."
      },
      "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": "original open index=5;open=[0, 1, 2, 3, 4];ACTUAL same-carrier complement=[]",
      "after": "complement open=是;nonempty proper subset=否;nontrivial clopen obstruction=否",
      "relation": "implies",
      "rule_id": "connected_every_actual_open_complement_not_selected_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原集合已開，再核補集開才是clopen；且必非空非全集才給分離。按拓撲原序列全體，不能由一個失敗候選推全域connected。",
        "en": "The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness."
      },
      "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": "ALL actual clopen sets=[[], [0, 1, 2, 3, 4]];ALL nontrivial clopen sets=[]",
      "after": "a nontrivial clopen S gives separation(S,X minus S);a separation by disjoint nonempty opens gives such a clopen S;actual connected iff no such set",
      "relation": "implies",
      "rule_id": "connected_clopen_separation_equivalence_full_finite_exhaustion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正向補集開且兩邊非空，互斥並聯為全集；反向分離兩開集互為補集，故其中一個是非平凡clopen。完整有限拓撲已逐項檢完，才可排除全部分離；不是未給圖的graph connectivity。",
        "en": "A proper nonempty clopen and its open complement separate the carrier.Conversely the two opens in a separation are complements,so either is nontrivial clopen.Exhausting the complete finite topology therefore excludes every separation.This is not connectivity of an unrelated graph."
      },
      "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": [
      [
        [],
        [
          0,
          1,
          2,
          3,
          4
        ]
      ],
      [],
      true,
      "exhaustive_complete_finite_topology"
    ]
  },
  "conclusion": {
    "canonical_value": "([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)",
    "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-20261014-advance-20261103-r05-connected-topological-space-foundations",
      "batch_path": "public/data/questions/revisions/ocme-reference-revision-20261017/batch-20261014-advance-20261103-r05-connected-topological-space-foundations/questions.jsonl",
      "batch_sha256": "3c891f48b55c54862066186c1c6685e645a1f09629e9f852bad8274c5a07215c",
      "original_line": 55,
      "original_line_sha256": "8db4d572479615f8803cd7e69014689c9987dd9d3b0dd008e0d060cfb36e7ce2",
      "question_compact_sha256": "1fddb1136c66d63ec774c3ade12638db26a79213f7e2dd557a4724d1aa09593e",
      "english_row_compact_sha256": "5d514f28be17ab3dc0a7335289375c5a48e4a3c5c18cc5f1476c5acee942af7a",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261103-r05-connected-topological-space-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261103-r05-connected-topological-space-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-connected-topological-space": {
        "sha256": "a3956f0120a382824010422c3639f93914810983c5f1dcf47b08e4f3b2d3d96f",
        "english_sha256": "805307688f6d22990517b8a23fe983a5aa40b37b3e8a5833f994f0828d791e90",
        "original_path": "public/data/mko/connected-topological-space.json",
        "english_path": "public/data/i18n/editions/mko-connected-topological-space.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c6f506c0e91ce68861b828c110bfee6002681c5a6e691f1ba5ae296376b30ad5",
    "original_candidate_sha256": "a763fa2228cf7f29438cc0b21b7b2df3bd1d55a7cadd7b74cc61ff5df725deb2",
    "original_candidate_line_sha256": "a763fa2228cf7f29438cc0b21b7b2df3bd1d55a7cadd7b74cc61ff5df725deb2"
  },
  "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-20261026-008-p34",
    "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": "4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3",
      "question_first_witness_sha256": "22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93"
    },
    "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": "4fafca72a73ca7b0eaf90904b8844b7c6ce9d2a26d98838df3d8e51a6441ffa7",
      "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."
  }
}