{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-subspace-topology-01",
  "mko_id": "mko-subspace-topology",
  "title": {
    "zh-Hant": "相對開集不等於母空間開集",
    "en": "Relative openness differs from ambient openness"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-subspace-topology-01",
    "source": {
      "mko_id": "mko-subspace-topology",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "84259c8587e1acd913189fcf6b17200828b88d36622591d8bc5a52b3455544e7",
      "english_sha256": "6f532d1a2d042136c6f62e7c763d13337ac407fea6f50219d476e51f4dea0198"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "sets",
      "topology",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "計算 Y={b,c} 的子空間拓撲與全部相對閉集。",
      "en": "Compute the subspace topology and all relative closed sets of Y={b,c}."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X={a,b,c}，tau_X={空集,{a},{a,b},X}，Y={b,c}。",
          "en": "X={a,b,c}, tau_X={empty,{a},{a,b},X}, Y={b,c}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "\\varnothing\\subset\\{a\\}\\subset\\{a,b\\}\\subset X",
        "after_tex": "\\tau_X\\text{ is a topology},\\quad Y\\subseteq X",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "母族是完整含兩端的有限鏈，聯交封閉；Y 確為子集。",
          "en": "The parent is a complete finite chain with endpoints and union/intersection closure; Y is indeed a subset."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不從欲求子空間族反推母族合法。",
            "en": "Parent validity is not inferred circularly from the desired subspace family."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S02",
        "uses": [
          "G1"
        ],
        "before_tex": "\\varnothing\\cap Y,\\ \\{a\\}\\cap Y",
        "after_tex": "\\varnothing,\\ \\varnothing",
        "relation": "equivalent",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "前兩個母開集的相交都空，保留重複直到正規化。",
          "en": "The first two ambient opens have empty intersections; keep the duplicate until normalization."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "空交集仍是合法相對開集。",
            "en": "An empty intersection is still a legal relative open."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S03",
        "uses": [
          "G1"
        ],
        "before_tex": "\\{a,b\\}\\cap Y,\\ X\\cap Y",
        "after_tex": "\\{b\\},\\ Y",
        "relation": "equivalent",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "後兩項給 {b} 與相對全集 Y。",
          "en": "The remaining intersections give {b} and the relative whole space Y."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "見證 {a,b} 不必包含於 Y。",
            "en": "The witnessing ambient open {a,b} need not lie inside Y."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S04",
        "uses": [
          "S02",
          "S03"
        ],
        "before_tex": "\\tau_Y=\\{U\\cap Y:U\\in\\tau_X\\}",
        "after_tex": "\\tau_Y=\\{\\varnothing,\\{b\\},Y\\}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "去除重複後列出完整相對開集。",
          "en": "After deduplication, list every relative open."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "四個母開集都已取交，因此沒有漏掉相對開集。",
            "en": "All four ambient opens were intersected, so no relative open is missing."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S05",
        "uses": [
          "S04"
        ],
        "before_tex": "\\varnothing\\subset\\{b\\}\\subset Y",
        "after_tex": "\\tau_Y\\text{ is a topology on }Y",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "再次以 Y 為全集核有限鏈的聯交公理。",
          "en": "With Y as the universe, verify the finite chain's union/intersection axioms."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04"
          ],
          "explanation": {
            "zh-Hant": "全集項是 Y，不是 X。",
            "en": "The whole-space member is Y, not X."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S06",
        "uses": [
          "S04"
        ],
        "before_tex": "\\{Y\\setminus V:V\\in\\tau_Y\\}",
        "after_tex": "\\mathcal F_Y=\\{Y,\\{c\\},\\varnothing\\}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "在 Y 中補集，得到全部相對閉集。",
          "en": "Complement within Y to obtain all relative closed sets."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04"
          ],
          "explanation": {
            "zh-Hant": "若錯在 X 補集，會得到不是 Y 子集的集合。",
            "en": "Complementing in X instead could produce sets not contained in Y."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S04"
        ],
        "before_tex": "\\{b\\}\\in\\tau_Y",
        "after_tex": "\\{b\\}\\notin\\tau_X",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "{b} 相對開但母空間不開，是兩個不同述詞。",
          "en": "{b} is relatively open but not ambient open; these are distinct predicates."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "母開集見證是 {a,b}，並非 {b} 自己。",
            "en": "The ambient-open witness is {a,b}, not {b} itself."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S08",
        "uses": [
          "S06",
          "S07"
        ],
        "before_tex": "Y\\setminus\\{c\\}=\\{b\\}\\in\\tau_Y",
        "after_tex": "\\{c\\}\\text{ is relatively closed};\\quad\\{b\\}\\text{ is not relatively closed}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "{c} 的相對補集開；{b} 的補集 {c} 不開。",
          "en": "The complement of {c} is relatively open; the complement {c} of {b} is not relatively open."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "相對閉性完全由完整相對拓撲核對。",
            "en": "Relative closedness is checked against the entire relative topology."
          }
        },
        "leaf_basis": "sets"
      }
    ],
    "result": {
      "zh-Hant": "tau_Y={空集,{b},Y}，相對閉集為 {空集,{c},Y}。",
      "en": "tau_Y={empty,{b},Y}; the relative closed sets are {empty,{c},Y}."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "見證母開集必須包含於 Y。",
          "en": "The witnessing ambient open must lie inside Y."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "{b} 的見證 {a,b} 不包含於 Y。",
          "en": "The witness {a,b} for {b} is not contained in Y."
        }
      },
      {
        "claim": {
          "zh-Hant": "{b} 在 X 中開。",
          "en": "{b} is open in X."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "未列在完整母拓撲。",
          "en": "It is absent from the complete ambient topology."
        }
      },
      {
        "claim": {
          "zh-Hant": "Y 的相對全集仍是 X。",
          "en": "The relative whole space is still X."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "相對拓撲的全集是 Y。",
          "en": "The relative topology has whole space Y."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "tau_Y={U∩Y:U∈tau_X}，重複與空交集正規化。relative-open未必ambient-open且不要求U⊆Y；relative-closed以Y內補集open或F∩Y、F ambient closed判定。全部拓撲公理以Y作相對全集。",
          "en": "The induced topology is {U∩Y:U∈tau_X}, normalizing duplicate and empty intersections. Relative opens need not be ambient open and their witnesses U need not lie inside Y. Relative closedness uses open complements inside Y, or intersections with ambient closed sets. Y is the relative universe in every topology obligation."
        },
        "source": {
          "mko_id": "mko-subspace-topology",
          "sha256": "84259c8587e1acd913189fcf6b17200828b88d36622591d8bc5a52b3455544e7",
          "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": "子空間拓撲為所有母開集與子空間載體的交；相對閉集是相對開集在子空間載體內的補集。",
          "en": "The subspace topology consists of intersections of ambient opens with the subspace carrier; relative closed sets are complements of relative opens inside that carrier."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整有限母拓撲，所有相對補集在 Y 內取。",
      "en": "A complete finite ambient topology; relative complements are inside Y."
    },
    "md_symbols": [
      [
        "Y",
        "子空間載體",
        "subspace carrier"
      ],
      [
        "\\tau_Y",
        "相對拓撲",
        "relative topology"
      ]
    ],
    "md_error_controls": [
      {
        "title": "交集全列 / List every intersection",
        "zh": "逐一遍歷母開集，再去除重複。",
        "en": "Traverse every ambient open before deduplicating."
      },
      {
        "title": "相對全集 / Relative universe",
        "zh": "相對補集與公理均固定使用 Y。",
        "en": "Relative complements and axioms consistently use Y."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
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    "original_source_sha256": "24419aa4e704355f75ed3b708e05ec2f2252aa18cea0d39bb65e43fd886167b6",
    "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."
  }
}