{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-topological-basis-01",
  "mko_id": "mko-topological-basis",
  "title": {
    "zh-Hant": "有限鏈的基底與全部生成開集",
    "en": "A finite-chain basis and every generated open"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-topological-basis-01",
    "source": {
      "mko_id": "mko-topological-basis",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "3aecebae96f5302a84e437770eaaec38b7992e61082720fa491302177ad43804",
      "english_sha256": "afbfa91d01de39a5536ec84bb9365435ac7fc714af5ea3f46ff7714bb03d0c78"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "sets",
      "topology",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "驗 B={{a},{a,b},X} 是指定拓撲的基底。",
      "en": "Verify B={{a},{a,b},X} is a basis for the specified topology."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X={a,b,c}，B={{a},{a,b},X}；目標 tau={空集,{a},{a,b},X}。",
          "en": "X={a,b,c}, B={{a},{a,b},X}; target tau={empty,{a},{a,b},X}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "B\\in\\mathcal B",
        "after_tex": "B\\subseteq X\\quad\\text{for all three members}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三個基底候選均在同一載體內。",
          "en": "Each of the three basis candidates lies in the same carrier."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不允許基底元素含載體以外的點。",
            "en": "No basis element may contain a point outside the carrier."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "\\bigcup\\mathcal B=\\{a\\}\\cup\\{a,b\\}\\cup X",
        "after_tex": "\\bigcup\\mathcal B=X",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "因 X 自身列在基底中，覆蓋條件成立。",
          "en": "Because X itself belongs to the family, coverage holds."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "覆蓋只是第一條基底公理，尚須交細化。",
            "en": "Coverage is only the first basis obligation; intersection refinement remains."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S03",
        "uses": [
          "G1"
        ],
        "before_tex": "B_1,B_2\\in\\mathcal B",
        "after_tex": "B_1\\cap B_2=\\min_{\\subseteq}(B_1,B_2)\\in\\mathcal B",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三個集合是包含鏈，六種無序可重複配對的交分別是較小那項。",
          "en": "The three members form an inclusion chain; each of the six unordered pairs with repetition intersects to its smaller member."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "鏈論法涵蓋全部配對，沒有未核例外。",
            "en": "The chain argument covers every pair without untested exceptions."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S04",
        "uses": [
          "S03"
        ],
        "before_tex": "x\\in B_1\\cap B_2",
        "after_tex": "B_3=B_1\\cap B_2,\\quad x\\in B_3\\subseteq B_1\\cap B_2",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "取交集本身作局部細化，因前一步已證它仍在基底。",
          "en": "Use the intersection itself as the local refinement, since the previous step puts it in the basis."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03"
          ],
          "explanation": {
            "zh-Hant": "不是要求所有基底一般都對交封閉；此例特別可直接取交。",
            "en": "This does not require arbitrary bases to be intersection-closed; this example permits that choice."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S05",
        "uses": [
          "G1"
        ],
        "before_tex": "\\mathcal T_{\\mathcal B}=\\{\\bigcup\\mathcal C:\\mathcal C\\subseteq\\mathcal B\\}",
        "after_tex": "\\mathcal T_{\\mathcal B}=\\{\\varnothing,\\{a\\},\\{a,b\\},X\\}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "八個子族的聯只有四個不同結果；空子族給空集，非空子族取最大項。",
          "en": "The eight subfamilies have four distinct unions; the empty subfamily gives empty and every nonempty one gives its largest member."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "列舉了全部聯而不是只列基底元素。",
            "en": "All unions were considered, not merely the basis members."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S06",
        "uses": [
          "S05"
        ],
        "before_tex": "\\mathcal T_{\\mathcal B}\\text{ is a finite inclusion chain with endpoints}",
        "after_tex": "\\mathcal T_{\\mathcal B}\\text{ is a topology}",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整四項族的聯交封閉性由鏈性得到。",
          "en": "The chain verifies union and intersection closure of the complete four-member family."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05"
          ],
          "explanation": {
            "zh-Hant": "目標族不是先被假定為拓撲，而是已核公理。",
            "en": "The target family was not merely assumed to be a topology; its axioms were checked."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S05"
        ],
        "before_tex": "\\mathcal B\\subseteq\\tau,\\quad\\mathcal T_{\\mathcal B}=\\tau",
        "after_tex": "\\mathcal B\\text{ is a basis for the supplied }\\tau",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個基底元素開且每個目標開集是基底聯，雙向生成責任都成立。",
          "en": "Every basis member is open and every target open is a basis union; both generation obligations hold."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "只核基底元素開仍不足，S05 提供反向生成。",
            "en": "Openness alone would be insufficient; S05 supplies generation of all opens."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S08",
        "uses": [
          "G1"
        ],
        "before_tex": "\\mathcal C=\\{\\{a,b\\},\\{b,c\\}\\}",
        "after_tex": "b\\in\\{a,b\\}\\cap\\{b,c\\},\\quad\\nexists C\\in\\mathcal C:\\ b\\in C\\subseteq\\{b\\}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "一個覆蓋 X 的族也可能沒有交細化，展示覆蓋不等於基底。",
          "en": "A family covering X may fail intersection refinement, demonstrating that coverage alone does not make a basis."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "失敗點 b 與全部兩個候選都明示，可窮盡拒絕。",
            "en": "The failed point b and both candidates are explicit, allowing exhaustive rejection."
          }
        },
        "leaf_basis": "sets"
      }
    ],
    "result": {
      "zh-Hant": "B 是指定 tau 的基底，生成恰好四個開集。",
      "en": "B is a basis of the supplied tau, generating exactly four opens."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "任何覆蓋都是基底。",
          "en": "Every cover is a basis."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "S08 的兩集合覆蓋卻在 b 無交細化。",
          "en": "The two-set cover in S08 lacks a refinement at b."
        }
      },
      {
        "claim": {
          "zh-Hant": "生成拓撲不必包含空集。",
          "en": "The generated topology need not contain empty."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "空聯必須列入。",
          "en": "The empty union must be included."
        }
      },
      {
        "claim": {
          "zh-Hant": "基底元素開已足夠證它生成既有拓撲。",
          "en": "Open basis candidates alone prove generation of an existing topology."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "還要每個既有開集都是它們的聯。",
          "en": "Every existing open must additionally be their union."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "每個basis element是X的子集且全部聯集覆蓋X；對x∈B1∩B2，存在B3使x∈B3⊆B1∩B2。若聲稱為既有拓撲的basis，還需basis elements皆開且每個open set為basis聯集。basis不可與subbasis混同。",
          "en": "Every basis element is a subset of X and their union covers X. If x lies in B1∩B2, some B3 contains x and lies inside that intersection. A basis for an existing topology additionally consists of open sets and generates every open set as a union. A basis must not be confused with a subbasis."
        },
        "source": {
          "mko_id": "mko-topological-basis",
          "sha256": "3aecebae96f5302a84e437770eaaec38b7992e61082720fa491302177ad43804",
          "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": "基底需覆蓋載體且每個交點有基底內局部細化；生成拓撲是所有子族之聯；既有拓撲的基底還需元素皆開且生成全部開集。",
          "en": "A basis covers the carrier and locally refines every intersection point; its generated topology consists of all subfamily unions; for an existing topology its members must be open and generate all opens."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整有限基底與拓撲；不混同子基底。",
      "en": "Complete finite basis and topology, distinct from a subbasis."
    },
    "md_symbols": [
      [
        "\\mathcal B",
        "基底候選",
        "candidate basis"
      ],
      [
        "\\mathcal T_{\\mathcal B}",
        "全部基底聯",
        "all basis unions"
      ]
    ],
    "md_error_controls": [
      {
        "title": "交點量詞 / Intersection-point quantifier",
        "zh": "每對及交內每點都須有細化，不能只核覆蓋。",
        "en": "Every pair and every intersection point needs a refinement, not just coverage."
      },
      {
        "title": "雙向生成 / Two-way generation",
        "zh": "核基底皆開，並核全部目標開集可生成。",
        "en": "Check basis openness and generation of every target open."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "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",
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    "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."
  }
}