{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-topological-connected-component-01",
  "mko_id": "mko-topological-connected-component",
  "title": {
    "zh-Hant": "完整子集表找包含極大的連通分量",
    "en": "Finding inclusion-maximal components from the complete subset table"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-topological-connected-component-01",
    "source": {
      "mko_id": "mko-topological-connected-component",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "a7a9f0548153129b46182a41ce190097c4d35b3765ef3987fbc9f4789b930b35",
      "english_sha256": "33a7bfd3c7917a20ed99b747cfbf51ae060b96e360efd93ea0e03d7f5fcdaf5d"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "sets",
      "topology",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "求 X={a,b,c}、tau={空集,{a},{b,c},X} 的全部連通分量。",
      "en": "Find every component of X={a,b,c}, tau={empty,{a},{b,c},X}."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X={a,b,c}, tau={空集,{a},{b,c},X}；空空間採連通。",
          "en": "X={a,b,c}, tau={empty,{a},{b,c},X}; the empty space is connected by convention."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "\\{a\\}\\cap\\{b,c\\}=\\varnothing,\\quad\\{a\\}\\cup\\{b,c\\}=X",
        "after_tex": "\\tau\\text{ is a topology}",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "除了兩個互補成員外，與空集全集運算均留在族內；全部聯交公理成立。",
          "en": "Apart from the complementary pair, operations with empty and whole stay in the family; every union/intersection axiom follows."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "四項清單完整。",
            "en": "The four-member list is complete."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S02",
        "uses": [
          "G1"
        ],
        "before_tex": "\\mathcal P(X)",
        "after_tex": "\\varnothing,\\{a\\},\\{b\\},\\{c\\},\\{a,b\\},\\{a,c\\},\\{b,c\\},X",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "列出全部八個子集，準備各自誘導拓撲。",
          "en": "List all eight subsets before computing their induced topologies."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "最大性必基於完整候選，不是只看較大集合。",
            "en": "Maximality must use all candidates, not just large-looking sets."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S03",
        "uses": [
          "S01",
          "S02"
        ],
        "before_tex": "Y=\\varnothing,\\{a\\},\\{b\\},\\{c\\}",
        "after_tex": "\\tau_\\varnothing=\\{\\varnothing\\},\\quad\\tau_{\\{p\\}}=\\{\\varnothing,\\{p\\}\\}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "四個小子集的誘導拓撲如列，都沒有非平凡分離。",
          "en": "The four small subsets have the listed induced topologies and no nontrivial separation."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "空集採已宣告慣例，單點只一個非空集合。",
            "en": "The declared convention handles empty, and a singleton has only one nonempty set."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S04",
        "uses": [
          "S01",
          "S02"
        ],
        "before_tex": "Y=\\{a,b\\},\\{a,c\\}",
        "after_tex": "\\tau_{\\{a,b\\}}=\\mathcal P(\\{a,b\\}),\\quad\\tau_{\\{a,c\\}}=\\mathcal P(\\{a,c\\})",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "相交母開集後得到兩點離散拓撲，各由兩單點分離，故不連通。",
          "en": "Ambient intersections give two-point discrete topologies, each separated by its two singletons and therefore disconnected."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "已列完整四項冪集描述，不以母空間點數推斷。",
            "en": "The complete four-set power sets are specified, rather than inferred from cardinality."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S05",
        "uses": [
          "S01",
          "S02"
        ],
        "before_tex": "Y=\\{b,c\\},\\ X",
        "after_tex": "\\tau_{\\{b,c\\}}=\\{\\varnothing,\\{b,c\\}\\};\\quad X=\\{a\\}\\sqcup\\{b,c\\}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "{b,c} 的誘導拓撲僅含空集與全集，因此連通；X 的兩個非空互補開集則明示一個分離。",
          "en": "The induced topology on {b,c} is indiscrete and connected; the two complementary nonempty opens explicitly separate X."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "至此八個子集的連通狀態全部決定。",
            "en": "Connectedness has now been decided for all eight subsets."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S06",
        "uses": [
          "S03",
          "S04",
          "S05"
        ],
        "before_tex": "\\mathcal C=\\{\\varnothing,\\{a\\},\\{b\\},\\{c\\},\\{b,c\\}\\}",
        "after_tex": "C_a=\\bigcup\\{A\\in\\mathcal C:a\\in A\\}=\\{a\\};\\quad C_b=C_c=\\{b,c\\}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "把含指定點的全部連通子集取聯，分別得到唯一包含極大候選。",
          "en": "Unite all connected subsets containing the given point, obtaining the unique inclusion-maximal candidates."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "{a} 基數小仍是極大，沒有被較大但不包含它的 {b,c} 取代。",
            "en": "The smaller {a} is maximal: the larger {b,c} does not contain it."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S07",
        "uses": [
          "S06"
        ],
        "before_tex": "\\{C_a,C_b\\}=\\{\\{a\\},\\{b,c\\}\\}",
        "after_tex": "C_a\\cap C_b=\\varnothing,\\quad C_a\\cup C_b=X",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩分量不同即不交，且完整覆蓋載體。",
          "en": "The distinct components are disjoint and cover the entire carrier."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06"
          ],
          "explanation": {
            "zh-Hant": "重複的 C_c=C_b 已正規化，不算第三分量。",
            "en": "The duplicate C_c=C_b is normalized, not counted as a third component."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07"
        ],
        "before_tex": "X\\setminus C_a=C_b\\in\\tau,\\quad X\\setminus C_b=C_a\\in\\tau",
        "after_tex": "C_a,C_b\\text{ both closed and open}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "本有限模型兩分量均閉且開；這不是一般空間分量必開的宣稱。",
          "en": "Both components are closed and open in this finite model; this does not assert general components are open."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "一般 Q 中分量是單點但不開；任意兩異有理數可用中間無理數切開，而每個有理鄰域含其他有理數。",
            "en": "In Q components are singletons but not open: an intermediate irrational separates any two rationals, while every rational neighborhood contains other rationals."
          }
        },
        "leaf_basis": "sets"
      }
    ],
    "result": {
      "zh-Hant": "分量為 {a} 與 {b,c}，它們分割 X，且在本有限空間均開閉。",
      "en": "The components {a} and {b,c} partition X and are clopen in this finite space."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "只有最大基數連通集才是分量。",
          "en": "Only a largest-cardinality connected set is a component."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "{a} 雖較小仍包含極大。",
          "en": "{a} is smaller but remains inclusion-maximal."
        }
      },
      {
        "claim": {
          "zh-Hant": "{a,b} 也是連通。",
          "en": "{a,b} is connected."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "其誘導拓撲是兩點離散，可分離。",
          "en": "Its induced two-point discrete topology has a separation."
        }
      },
      {
        "claim": {
          "zh-Hant": "一般空間的分量都開。",
          "en": "Components are open in every space."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "Q 的單點分量不開，S08 已給分離與非開理由。",
          "en": "The singleton components of Q are not open, as justified at S08."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "finite topology families完整枚舉subspace topologies與所有connected subsets；含x的connected subsets之union給唯一inclusion-maximal component。再核components兩兩相等或不交、覆蓋X並在指定finite model中closed。有限拓撲因只有有限多components而使每個component同時open，這是有限特例；一般空間的component不必open，Q的通常subspace topology中singletons是components但不open。不得把最大基數、finite-graph connected component或其他結構混同。",
          "en": "Finite families enumerate every induced subspace and connected subset. The union of all connected subsets containing x is its unique inclusion-maximal component. Then check that the components are pairwise equal or disjoint, cover X and are closed in the stated finite model. Finitely many closed components also make each component open, a finite-specific conclusion. In a general space components need not be open: in the rational subspace of the real line they are singletons and are not open. Do not substitute globally largest cardinality, graph components or a different structure on the same carrier."
        },
        "source": {
          "mko_id": "mko-topological-connected-component",
          "sha256": "a7a9f0548153129b46182a41ce190097c4d35b3765ef3987fbc9f4789b930b35",
          "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": "子空間開集為母開集之交；連通子集無非平凡相對開分離；含 x 的分量是全部含 x 連通子集之聯，亦為包含極大連通子集。",
          "en": "Subspace opens are ambient-open intersections; a connected subset has no nontrivial relative-open separation; the component of x is the union of all connected subsets containing x, equivalently the inclusion-maximal connected subset."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "逐一使用誘導拓撲，不借用圖，有限 clopen 結論不外推。",
      "en": "Use each induced topology, not a graph; do not extrapolate finite clopen components."
    },
    "md_symbols": [
      [
        "C_x",
        "含 x 的連通分量",
        "component containing x"
      ],
      [
        "\\mathcal C",
        "全部連通子集",
        "all connected subsets"
      ]
    ],
    "md_error_controls": [
      {
        "title": "完整子集表 / Complete subset table",
        "zh": "先核八個誘導子空間，再判包含極大。",
        "en": "Check all eight induced subspaces before inclusion-maximality."
      },
      {
        "title": "不混最大 / Maximal is not largest",
        "zh": "極大是不能被真包含於另一連通子集，不是基數最大。",
        "en": "Maximal means no proper connected superset, not largest cardinality."
      }
    ],
    "md_open_items": []
  },
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