{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-closure-and-interior-01",
  "mko_id": "mko-closure-and-interior",
  "title": {
    "zh-Hant": "在鏈狀三點拓撲求閉包與內部",
    "en": "Closure and interior in a three-point chain topology"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-closure-and-interior-01",
    "source": {
      "mko_id": "mko-closure-and-interior",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "72aef7da5935f151015d5eecaa4d9cdde8226564638885f222596bf4ef2433e4",
      "english_sha256": "8dc1d3fac60ca206027663c117e374595aff835c923f99ec01bdb7f6451fd1d2"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "sets",
      "topology",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "求 A={b} 的閉包及內部。",
      "en": "Find the closure and interior of A={b}."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X={a,b,c}，完整候選族 tau={空集,{a},{a,b},X}，A={b}。",
          "en": "X={a,b,c}, the entire candidate family is tau={empty,{a},{a,b},X}, and A={b}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "\\tau=\\{\\varnothing,\\{a\\},\\{a,b\\},X\\}",
        "after_tex": "\\varnothing,X\\in\\tau",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先核所有成員都是 X 的子集且包含兩個必要端點。",
          "en": "Every listed member is a subset of X, and both required endpoint sets are present."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "只核端點不足以完成公理；後面仍要核交與聯。",
            "en": "Endpoint membership alone is insufficient; intersections and unions still need checking."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "\\varnothing\\subset\\{a\\}\\subset\\{a,b\\}\\subset X",
        "after_tex": "U\\cup V=\\max_{\\subseteq}(U,V),\\quad U\\cap V=\\min_{\\subseteq}(U,V)",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "族是有限包含鏈，非空子族的聯交分別是最大與最小成員。空聯為空集，空交為 X，故全部公理成立。",
          "en": "The family is a finite inclusion chain. For every nonempty subfamily, its union is its largest member and its intersection is its smallest member. The empty union is the empty set, and the empty intersection is X. All topology axioms follow."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "這是列出的整族的鏈性，不是只核一對集合。",
            "en": "The chain argument covers the complete listed family, not one tested pair."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "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": "逐一取相對 X 的補集，得到全部閉集，不遺漏空集。",
          "en": "Take each complement relative to X to obtain every closed set, including the empty set."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "把四個閉集再取補集恰好回到原拓撲，確認全集綁定。",
            "en": "Complementing the four closed sets returns exactly the topology and checks the ambient carrier."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03"
        ],
        "before_tex": "A=\\{b\\},\\quad A\\subseteq F\\in\\mathcal F",
        "after_tex": "\\{F\\in\\mathcal F:A\\subseteq F\\}=\\{\\{b,c\\},X\\}",
        "relation": "equivalent",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "空集與 {c} 都漏掉 b，所以只有另外兩個閉超集。",
          "en": "Empty and {c} miss b; the other two closed sets are precisely the closed supersets."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "兩個入選集合都含 b，兩個落選集合都不含，已窮盡四項。",
            "en": "Both selected sets contain b and both rejected sets do not; all four have been exhausted."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S05",
        "uses": [
          "S04"
        ],
        "before_tex": "\\overline A=\\bigcap\\{F\\text{ closed}:A\\subseteq F\\}",
        "after_tex": "\\overline A=\\{b,c\\}\\cap X=\\{b,c\\}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "閉包是所有閉超集的交，不是任挑其中一個。",
          "en": "The closure is the intersection of all closed supersets, not an arbitrary selected superset."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04"
          ],
          "explanation": {
            "zh-Hant": "{b,c} 是閉且含 A；任何閉超集都含它，核到最小性。",
            "en": "The result is closed and contains A; every closed superset contains it, verifying minimality."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "U\\in\\tau,\\quad U\\subseteq\\{b\\}",
        "after_tex": "\\{U\\in\\tau:U\\subseteq A\\}=\\{\\varnothing\\}",
        "relation": "equivalent",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個非空開集都含 a，因此不能包含在 {b} 中。",
          "en": "Every nonempty open set contains a, so none lies in {b}."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "核對全部三個非空開集而不是只看最小開集的名稱。",
            "en": "All three nonempty opens have been checked, rather than inferred from a label."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S07",
        "uses": [
          "S06"
        ],
        "before_tex": "A^\\circ=\\bigcup\\{U\\in\\tau:U\\subseteq A\\}",
        "after_tex": "A^\\circ=\\varnothing",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "內部是全部開子集的聯；這裡唯一項是空集。",
          "en": "The interior unites all open subsets; the only such member is empty."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06"
          ],
          "explanation": {
            "zh-Hant": "空集確實開且在 A 中；沒有較大的開子集可加入。",
            "en": "Empty is open and contained in A, and no larger open subset was omitted."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S08",
        "uses": [
          "S05",
          "S07"
        ],
        "before_tex": "A^\\circ\\subseteq A\\subseteq\\overline A",
        "after_tex": "\\varnothing\\subseteq\\{b\\}\\subseteq\\{b,c\\}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "將計算結果代回兩個基本包含關係，並記住換拓撲會換答案。",
          "en": "Substitute the results into both basic inclusions; a different topology can give different answers."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "包含檢查只是結果核對；真正的閉包與內部由先前完整交聯給出。",
            "en": "The inclusions are consistency checks; the exhaustive intersection and union establish the actual constructions."
          }
        },
        "leaf_basis": "sets"
      }
    ],
    "result": {
      "zh-Hant": "在指定拓撲，閉包 {b,c}、內部空集。空集與 X 的閉包及內部分別仍是自己。",
      "en": "In this topology the closure is {b,c} and the interior is empty. Empty and X each equal their own closure and interior."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "閉包是 X，因為 X 閉且含 A。",
          "en": "The closure is X because X is closed and contains A."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "漏掉更小閉超集 {b,c}。",
          "en": "The smaller closed superset {b,c} was omitted."
        }
      },
      {
        "claim": {
          "zh-Hant": "{b} 的內部是 {b}。",
          "en": "The interior of {b} is {b}."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "{b} 不在完整開集族。",
          "en": "{b} is absent from the complete open family."
        }
      },
      {
        "claim": {
          "zh-Hant": "上述答案對 X 上所有拓撲都一樣。",
          "en": "These answers hold for every topology on X."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "離散拓撲使 {b} 自己開且閉。",
          "en": "In the discrete topology {b} itself is open and closed."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "題目先完整驗證有限拓撲，再由全部閉超集或開子集精確求閉包與內部。空集、全空間及拓撲相對性均保留，不暗中使用度量。",
          "en": "First validate the complete finite topology, then calculate closure and interior exactly from all closed supersets or open subsets. Retain the empty-set and whole-space boundaries and dependence on the topology; do not silently introduce a metric."
        },
        "source": {
          "mko_id": "mko-closure-and-interior",
          "sha256": "72aef7da5935f151015d5eecaa4d9cdde8226564638885f222596bf4ef2433e4",
          "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": "閉集是開集的相對補集；閉包為全部閉超集之交，內部為全部開子集之聯。",
          "en": "Closed sets are relative complements of opens; closure intersects all closed supersets and interior unites all open subsets."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整有限拓撲的逐集計算；沒有指定度量。",
      "en": "An exhaustive finite-topology calculation; no metric is supplied."
    },
    "md_symbols": [
      [
        "\\tau",
        "完整開集族",
        "complete open family"
      ],
      [
        "\\overline A",
        "閉包",
        "closure"
      ],
      [
        "A^\\circ",
        "內部",
        "interior"
      ]
    ],
    "md_error_controls": [
      {
        "title": "全族責任 / Whole-family obligation",
        "zh": "先列全部閉超集再取交，先列全部開子集再取聯。",
        "en": "List every closed superset before intersecting and every open subset before uniting."
      },
      {
        "title": "全集綁定 / Ambient binding",
        "zh": "補集均相對 X，不相對 A。",
        "en": "All complements are relative to X, not A."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "original_source_sha256": "2fa41873c0ddcc9253a540235a90fa71fd13d0554b298df20b082b0267e4e54a",
    "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": [
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    "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."
  }
}