{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-topological-boundary-01",
  "mko_id": "mko-topological-boundary",
  "title": {
    "zh-Hant": "邊界與補集的精確對照",
    "en": "Exact boundary comparison with the complement"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-topological-boundary-01",
    "source": {
      "mko_id": "mko-topological-boundary",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "31f941e14ac65e7366835ceef8264cd169c8739bdaa6e04182c4f7b160ec8a13",
      "english_sha256": "c6431291562b0578aba8220926ac7692cbde83c7b924a496bb5e74c445028500"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "sets",
      "topology",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "計算鏈狀拓撲中 A={b} 與其補集的邊界。",
      "en": "Compute the boundaries of A={b} and its complement in a chain topology."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X={a,b,c}，tau={空集,{a},{a,b},X}，A={b}，D=X\\A={a,c}。",
          "en": "X={a,b,c}, tau={empty,{a},{a,b},X}, A={b}, D=X minus A={a,c}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "\\varnothing\\subset\\{a\\}\\subset\\{a,b\\}\\subset X",
        "after_tex": "\\tau\\text{ is a topology}",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "包含鏈中任意非空子族的聯為最大項、交為最小項；空聯和空交也在族內。",
          "en": "In the inclusion chain, unions and intersections of nonempty subfamilies are their largest and smallest members; empty union/intersection are included."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "此族完整列出四項，鏈性足以核全部公理。",
            "en": "All four members are supplied, so the chain argument checks all axioms."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "\\mathcal F=\\{X\\setminus U:U\\in\\tau\\}",
        "after_tex": "\\mathcal F=\\{X,\\{b,c\\},\\{c\\},\\varnothing\\}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "相對 X 逐項補集得到全部閉集。",
          "en": "Complement each open relative to X to obtain all closed sets."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "每個補集再補一次恰好回原開集。",
            "en": "Complementing twice returns the corresponding original open."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "F\\supseteq A,\\quad F\\in\\mathcal F",
        "after_tex": "\\overline A=X\\cap\\{b,c\\}=\\{b,c\\}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "含 b 的閉集恰為 X 與 {b,c}。",
          "en": "The closed sets containing b are exactly X and {b,c}."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "{c} 與空集不含 b，故閉超集枚舉完整。",
            "en": "{c} and empty do not contain b; the enumeration is complete."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "U\\subseteq A,\\quad U\\in\\tau",
        "after_tex": "A^\\circ=\\bigcup\\{\\varnothing\\}=\\varnothing",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "所有非空開集含 a，故沒有非空開子集落在 A 中。",
          "en": "All nonempty opens contain a, so no nonempty open lies inside A."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "並未把集合自身誤當成開集。",
            "en": "A itself has not been silently assumed open."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S05",
        "uses": [
          "S03",
          "S04"
        ],
        "before_tex": "\\partial A=\\overline A\\setminus A^\\circ",
        "after_tex": "\\partial A=\\{b,c\\}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "直接按邊界定義作差集。",
          "en": "Apply the closure-minus-interior definition directly."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "結果不必是 A 的子集；c 在邊界但不在 A。",
            "en": "The boundary need not lie in A: c is a boundary point outside A."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S01",
          "S02"
        ],
        "before_tex": "D=\\{a,c\\}",
        "after_tex": "\\overline D=X,\\quad D^\\circ=\\{a\\}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "含 a,c 的唯一閉集為 X；包含於 D 的開集是空集與 {a}。",
          "en": "X is the only closed set containing a,c; empty and {a} are exactly the opens contained in D."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "對 D 重新完整列舉，不由圖形猜對稱。",
            "en": "Re-enumerate for D instead of guessing symmetry from a drawing."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S07",
        "uses": [
          "S05",
          "S06"
        ],
        "before_tex": "\\partial D=\\overline D\\setminus D^\\circ",
        "after_tex": "\\partial D=\\{b,c\\}=\\partial A=\\overline A\\cap\\overline D",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "補集邊界及雙閉包交集都給同一結果。",
          "en": "The complement boundary and the intersection of the two closures give the same result."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "三個集合等式逐元素核對，只聲稱此明示例子。",
            "en": "Check all three equalities elementwise, claiming this explicit instance only."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S08",
        "uses": [
          "S01",
          "S02"
        ],
        "before_tex": "\\overline\\varnothing=\\varnothing,\\ \\varnothing^\\circ=\\varnothing,\\ \\overline X=X=X^\\circ",
        "after_tex": "\\partial\\varnothing=\\partial X=\\varnothing",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "空集與全集的邊界都空，避免把全空間外緣當未定義的幾何概念。",
          "en": "Empty and whole space have empty boundary; no undefined geometric outer edge is used."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "每個閉包與內部均可由已列族直接核對。",
            "en": "Each closure and interior follows directly from the listed families."
          }
        },
        "leaf_basis": "sets"
      }
    ],
    "result": {
      "zh-Hant": "兩個互補集合的邊界皆 {b,c}；空集及 X 邊界皆空。",
      "en": "Both complementary sets have boundary {b,c}; empty and X have empty boundaries."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "邊界必包含於 A。",
          "en": "The boundary must lie in A."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "c 屬邊界卻不屬 A。",
          "en": "c is on the boundary but not in A."
        }
      },
      {
        "claim": {
          "zh-Hant": "X 的邊界是 X。",
          "en": "The boundary of X is X."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "X 的閉包及內部均 X，作差後為空。",
          "en": "Closure and interior of X are both X, giving an empty difference."
        }
      },
      {
        "claim": {
          "zh-Hant": "此結果由三點之間的距離決定。",
          "en": "The result is determined by distances between the points."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "未指定度量，答案來自所列拓撲。",
          "en": "No metric is supplied; the result comes from the listed topology."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "完整驗有限拓撲後精確計算closure與interior。保留boundary(A)=boundary(complement A)、空集與全空間邊界，以及拓撲相對性。neighborhood contact若使用必須枚舉全部open neighborhoods。",
          "en": "After validating the complete finite topology, calculate closure and interior exactly. Retain boundary(A)=boundary(complement A), the empty/whole-set boundaries and dependence on the topology. If a neighborhood-contact criterion is used, every open neighborhood must be considered."
        },
        "source": {
          "mko_id": "mko-topological-boundary",
          "sha256": "31f941e14ac65e7366835ceef8264cd169c8739bdaa6e04182c4f7b160ec8a13",
          "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": "閉集為開集補集，閉包取全部閉超集交，內部取全部開子集聯；邊界為閉包減內部。",
          "en": "Closed sets complement opens, closure intersects all closed supersets, interior unites all open subsets, and boundary is closure minus interior."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整有限開閉集計算，未使用度量或圖形。",
      "en": "Exhaustive finite open/closed-set calculation, without a metric or drawing."
    },
    "md_symbols": [
      [
        "\\partial A",
        "拓撲邊界",
        "topological boundary"
      ],
      [
        "D",
        "相對 X 的補集",
        "complement relative to X"
      ]
    ],
    "md_error_controls": [
      {
        "title": "兩側獨立計算 / Compute both sides",
        "zh": "對 A 及補集分別列完整開子集與閉超集。",
        "en": "List all open subsets and closed supersets for A and its complement independently."
      },
      {
        "title": "不換全集 / Keep the carrier",
        "zh": "差集和補集固定相對 X。",
        "en": "Differences and complements use the same X."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "21eba37c5fc910925cb200b0b393070d907805f2655a1be31714240ec9c5add6",
    "original_source_sha256": "21eba37c5fc910925cb200b0b393070d907805f2655a1be31714240ec9c5add6",
    "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."
  }
}