{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261117-r07-topological-connected-component-0051",
  "pilot_number": "R0052-26576",
  "number": 26576,
  "family_id": "qf-mod48-component-partition",
  "title": {
    "zh-Hant": "第 26576 題：拓撲連通分支",
    "en": "Question 26576: Finite Topological Components Partition the Space and Are Clopen"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261117-r07-topological-connected-component-0051",
    "batch_id": "batch-20261015-advance-20261117-r07-topological-connected-component-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-topological-connected-component"
    ],
    "title_zh": "拓撲連通分支",
    "stem_zh": "subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 1], [2, 3, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)",
      "explanation_zh": "Wave6第十一層核對=([[0, 1], [2, 3, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "拓撲連通分支"
      ]
    },
    "variant": {
      "family_id": "qf-mod48-component-partition",
      "seed": 202611177051,
      "parameters": {
        "points": [
          0,
          1,
          2,
          3,
          4,
          5
        ],
        "topology": [
          [],
          [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          [
            0,
            1
          ],
          [
            2,
            3,
            4,
            5
          ]
        ],
        "domain": "Complete_finite_topological_components",
        "operation": "component-partition",
        "semantic_template": "component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261117-r07-topological-connected-component-0051",
    "batch_id": "batch-20261015-advance-20261117-r07-topological-connected-component-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "aed96f958c33dbb23017f73646b5e067eeb196869df9604aae4a847dc109d7ea",
      "family_id": "qf-mod48-component-partition",
      "seed": 202611177051,
      "semantic_template": "component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}"
    },
    "fields": {
      "title": "Finite Topological Components Partition the Space and Are Clopen",
      "stem": "Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1], [2, 3, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([[0, 1], [2, 3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)",
        "explanation": "Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1], [2, 3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."
      },
      "skills": [
        "Finite Topological Components Partition the Space and Are Clopen"
      ],
      "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": "subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1], [2, 3, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [component-partition]",
      "after": "derived = [[components, disjoint, cover, closed_flags, open_flags, valid, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "直接列出此操作及mode的完整來源欄序，不需JavaScript；布林、字串與null角色保持區別，不補欄或以答案形狀猜介面。",
        "en": "List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。",
          "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [component ORIGINAL space, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1], [2, 3, 4, 5]], 否]",
      "after": "derived = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1], [2, 3, 4, 5]], [[0, 0, 0, 0, 是, 是], [0, 1, 63, 0, 是, 是], [0, 2, 3, 0, 是, 是], [0, 3, 60, 0, 是, 是], [1, 0, 63, 0, 是, 是], [1, 1, 63, 63, 是, 是], [1, 2, 63, 3, 是, 是], [1, 3, 63, 60, 是, 是], [2, 0, 3, 0, 是, 是], [2, 1, 63, 3, 是, 是], [2, 2, 3, 3, 是, 是], [2, 3, 63, 0, 是, 是], [3, 0, 60, 0, 是, 是], [3, 1, 63, 60, 是, 是], [3, 2, 63, 0, 是, 是], [3, 3, 60, 60, 是, 是]]]",
      "relation": "implies",
      "rule_id": "Topology_full_typed_carrier_and_ALL_pair_axioms",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留JSON型別，set與open族依來源English序。逐全部open pair驗union及intersection，含空集與全集；有限族的二元閉包給任意union閉包，不由抽樣認定topology。",
        "en": "Retain JSON types and English source ordering of sets and open families.Check union/intersection for every open pair and empty/full sets;in a finite family binary closure gives arbitrary-union closure,not sampled qualification."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。",
          "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1], [2, 3, 4, 5]]]",
      "after": "derived = [[[1, [0], [[], [0]], null, 是], [2, [1], [[], [1]], null, 是], [3, [0, 1], [[], [0, 1]], null, 是], [4, [2], [[], [2]], null, 是], [5, [0, 2], [[], [0, 2], [0], [2]], [0], 否], [6, [1, 2], [[], [1, 2], [1], [2]], [1], 否], [7, [0, 1, 2], [[], [0, 1, 2], [0, 1], [2]], [0, 1], 否], [8, [3], [[], [3]], null, 是], [9, [0, 3], [[], [0, 3], [0], [3]], [0], 否], [10, [1, 3], [[], [1, 3], [1], [3]], [1], 否], [11, [0, 1, 3], [[], [0, 1, 3], [0, 1], [3]], [0, 1], 否], [12, [2, 3], [[], [2, 3]], null, 是], [13, [0, 2, 3], [[], [0, 2, 3], [0], [2, 3]], [0], 否], [14, [1, 2, 3], [[], [1, 2, 3], [1], [2, 3]], [1], 否], [15, [0, 1, 2, 3], [[], [0, 1, 2, 3], [0, 1], [2, 3]], [0, 1], 否], [16, [4], [[], [4]], null, 是], [17, [0, 4], [[], [0, 4], [0], [4]], [0], 否], [18, [1, 4], [[], [1, 4], [1], [4]], [1], 否], [19, [0, 1, 4], [[], [0, 1, 4], [0, 1], [4]], [0, 1], 否], [20, [2, 4], [[], [2, 4]], null, 是], [21, [0, 2, 4], [[], [0, 2, 4], [0], [2, 4]], [0], 否], [22, [1, 2, 4], [[], [1, 2, 4], [1], [2, 4]], [1], 否], [23, [0, 1, 2, 4], [[], [0, 1, 2, 4], [0, 1], [2, 4]], [0, 1], 否], [24, [3, 4], [[], [3, 4]], null, 是], [25, [0, 3, 4], [[], [0, 3, 4], [0], [3, 4]], [0], 否], [26, [1, 3, 4], [[], [1, 3, 4], [1], [3, 4]], [1], 否], [27, [0, 1, 3, 4], [[], [0, 1, 3, 4], [0, 1], [3, 4]], [0, 1], 否], [28, [2, 3, 4], [[], [2, 3, 4]], null, 是], [29, [0, 2, 3, 4], [[], [0, 2, 3, 4], [0], [2, 3, 4]], [0], 否], [30, [1, 2, 3, 4], [[], [1, 2, 3, 4], [1], [2, 3, 4]], [1], 否], [31, [0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0, 1], [2, 3, 4]], [0, 1], 否], [32, [5], [[], [5]], null, 是], [33, [0, 5], [[], [0, 5], [0], [5]], [0], 否], [34, [1, 5], [[], [1, 5], [1], [5]], [1], 否], [35, [0, 1, 5], [[], [0, 1, 5], [0, 1], [5]], [0, 1], 否], [36, [2, 5], [[], [2, 5]], null, 是], [37, [0, 2, 5], [[], [0, 2, 5], [0], [2, 5]], [0], 否], [38, [1, 2, 5], [[], [1, 2, 5], [1], [2, 5]], [1], 否], [39, [0, 1, 2, 5], [[], [0, 1, 2, 5], [0, 1], [2, 5]], [0, 1], 否], [40, [3, 5], [[], [3, 5]], null, 是], [41, [0, 3, 5], [[], [0, 3, 5], [0], [3, 5]], [0], 否], [42, [1, 3, 5], [[], [1, 3, 5], [1], [3, 5]], [1], 否], [43, [0, 1, 3, 5], [[], [0, 1, 3, 5], [0, 1], [3, 5]], [0, 1], 否], [44, [2, 3, 5], [[], [2, 3, 5]], null, 是], [45, [0, 2, 3, 5], [[], [0, 2, 3, 5], [0], [2, 3, 5]], [0], 否], [46, [1, 2, 3, 5], [[], [1, 2, 3, 5], [1], [2, 3, 5]], [1], 否], [47, [0, 1, 2, 3, 5], [[], [0, 1, 2, 3, 5], [0, 1], [2, 3, 5]], [0, 1], 否], [48, [4, 5], [[], [4, 5]], null, 是], [49, [0, 4, 5], [[], [0, 4, 5], [0], [4, 5]], [0], 否], [50, [1, 4, 5], [[], [1, 4, 5], [1], [4, 5]], [1], 否], [51, [0, 1, 4, 5], [[], [0, 1, 4, 5], [0, 1], [4, 5]], [0, 1], 否], [52, [2, 4, 5], [[], [2, 4, 5]], null, 是], [53, [0, 2, 4, 5], [[], [0, 2, 4, 5], [0], [2, 4, 5]], [0], 否], [54, [1, 2, 4, 5], [[], [1, 2, 4, 5], [1], [2, 4, 5]], [1], 否], [55, [0, 1, 2, 4, 5], [[], [0, 1, 2, 4, 5], [0, 1], [2, 4, 5]], [0, 1], 否], [56, [3, 4, 5], [[], [3, 4, 5]], null, 是], [57, [0, 3, 4, 5], [[], [0, 3, 4, 5], [0], [3, 4, 5]], [0], 否], [58, [1, 3, 4, 5], [[], [1, 3, 4, 5], [1], [3, 4, 5]], [1], 否], [59, [0, 1, 3, 4, 5], [[], [0, 1, 3, 4, 5], [0, 1], [3, 4, 5]], [0, 1], 否], [60, [2, 3, 4, 5], [[], [2, 3, 4, 5]], null, 是], [61, [0, 2, 3, 4, 5], [[], [0, 2, 3, 4, 5], [0], [2, 3, 4, 5]], [0], 否], [62, [1, 2, 3, 4, 5], [[], [1, 2, 3, 4, 5], [1], [2, 3, 4, 5]], [1], 否], [63, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1], [2, 3, 4, 5]], [0, 1], 否]], [[0], [1], [0, 1], [2], [3], [2, 3], [4], [2, 4], [3, 4], [2, 3, 4], [5], [2, 5], [3, 5], [2, 3, 5], [4, 5], [2, 4, 5], [3, 4, 5], [2, 3, 4, 5]]]",
      "relation": "implies",
      "rule_id": "Component_ALL_nonempty_masks_induced_topology_no_clopen_separation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個非空subset都用O∩A生成全部subspace opens，再找proper非空part及其相對補集同開的separation。不存在者才連通；保numeric mask序，singleton自然連通。",
        "en": "For every nonempty subset generate all subspace opens O intersect A,then test every proper nonempty part and its relative complement for an open separation.Absence qualifies connectedness;retain numeric-mask order,including singletons."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。",
          "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [[[0], [1], [0, 1], [2], [3], [2, 3], [4], [2, 4], [3, 4], [2, 3, 4], [5], [2, 5], [3, 5], [2, 3, 5], [4, 5], [2, 4, 5], [3, 4, 5], [2, 3, 4, 5]]]",
      "after": "derived = [[[0, 1], [2, 3, 4, 5]], 是, 是, [是, 是], [是, 是]]",
      "relation": "implies",
      "rule_id": "Component_inclusion_MAXIMAL_partition_finite_clopen",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從完整連通族按包含關係取maximal，不取最大基數。兩分支若交非空，其聯集連通使maximal性迫相同；每點singleton被包含所以覆蓋。分支closure仍連通故closed；有限多closed分支使每個補集closed，因此本有限空間亦open。",
        "en": "Select inclusion-maximal members of the complete connected family,not maximum cardinality.Intersecting components have connected union and maximality makes them equal;singletons give coverage.The closure of a connected set remains connected,so components are closed;finitely many closed components make every complement closed,hence each is open here."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。",
          "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        [
          0,
          1
        ],
        [
          2,
          3,
          4,
          5
        ]
      ],
      true,
      true,
      [
        true,
        true
      ],
      [
        true,
        true
      ],
      true,
      "finite_components_partition_and_are_clopen_finite_specific"
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 1], [2, 3, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)",
    "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-20261015-advance-20261117-r07-topological-connected-component-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261117-r07-topological-connected-component-foundations/questions.jsonl",
      "batch_sha256": "be9c78ee07ac57cfd398062cd24fa5cb4393ebafab7604a70e76ff63ba6e223d",
      "original_line": 51,
      "original_line_sha256": "88232702f3dc5fac6e001069b96a726abcf3e539acdb47c229b034f54d5f1d58",
      "question_compact_sha256": "aed96f958c33dbb23017f73646b5e067eeb196869df9604aae4a847dc109d7ea",
      "english_row_compact_sha256": "b2088e7c83cb4038692a4f823bc6f9c080481b38f5fa1a0b3e0292db70945714",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261117-r07-topological-connected-component-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261117-r07-topological-connected-component-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-topological-connected-component": {
        "sha256": "a7a9f0548153129b46182a41ce190097c4d35b3765ef3987fbc9f4789b930b35",
        "english_sha256": "33a7bfd3c7917a20ed99b747cfbf51ae060b96e360efd93ea0e03d7f5fcdaf5d",
        "original_path": "public/data/mko/topological-connected-component.json",
        "english_path": "public/data/i18n/editions/mko-topological-connected-component.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "a3def7368821a4d3dc7757fefd3c4a099713f8de8fb8e6f3434025e9d7b5a6d0",
    "original_candidate_sha256": "5a790e81f3cd34ee983b686aeef524a0ae5a1b0c654335ae500e0a4d60e7d41c",
    "original_candidate_line_sha256": "5a790e81f3cd34ee983b686aeef524a0ae5a1b0c654335ae500e0a4d60e7d41c",
    "mko_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c",
      "previous_english_sha256": {
        "mko-topological-connected-component": "ef1b70543c0fcf13716e8c687e2ddf1e1596c3fa0990cb5cf0a9dfda5d702a7c"
      }
    }
  },
  "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-20261026-010-p53",
    "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": "729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252",
      "question_first_witness_sha256": "d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 4,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "ecc6f659d4e8f830a17a8543f886fafd6cae004b881b56a00911395dfd40e3ce",
      "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."
  }
}