{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-connected-topological-space-01",
  "mko_id": "mko-connected-topological-space",
  "title": {
    "zh-Hant": "排除全部非平凡開閉集",
    "en": "Excluding every nontrivial clopen set"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-connected-topological-space-01",
    "source": {
      "mko_id": "mko-connected-topological-space",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "a3956f0120a382824010422c3639f93914810983c5f1dcf47b08e4f3b2d3d96f",
      "english_sha256": "805307688f6d22990517b8a23fe983a5aa40b37b3e8a5833f994f0828d791e90"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "sets",
      "topology",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "判斷三點鏈狀拓撲是否連通。",
      "en": "Decide connectedness of a three-point chain topology."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X={a,b,c}，tau={空集,{a},{a,b},X}；連通指不存在非空不交開集對覆蓋 X。",
          "en": "X={a,b,c}, tau={empty,{a},{a,b},X}; connected means no two disjoint nonempty opens cover X."
        }
      }
    ],
    "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": "有限鏈的非空聯交各取極端項，空聯與全集也在族中。",
          "en": "Nonempty unions/intersections of this finite chain take extremal members; empty and whole sets are present."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "先驗合法拓撲才判連通。",
            "en": "A valid topology is established before classifying connectedness."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S02",
        "uses": [
          "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": "補集枚舉全部閉集。",
          "en": "Enumerate all closed sets by complementation."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01"
          ],
          "explanation": {
            "zh-Hant": "補集固定相對 X。",
            "en": "Complements are relative to the fixed X."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S03",
        "uses": [
          "S01",
          "S02"
        ],
        "before_tex": "C\\in\\tau\\cap\\mathcal F",
        "after_tex": "\\tau\\cap\\mathcal F=\\{\\varnothing,X\\}",
        "relation": "equivalent",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "{a} 與 {a,b} 的補集都不開，沒有其他非平凡開閉集。",
          "en": "The complements of {a} and {a,b} are not open, leaving no nontrivial clopen set."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "四個開集均已對照完整閉集表。",
            "en": "All four opens have been compared with the full closed list."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S04",
        "uses": [
          "G1"
        ],
        "before_tex": "X=U\\cup V,\\ U,V\\ne\\varnothing,\\ U\\cap V=\\varnothing",
        "after_tex": "V=X\\setminus U",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "若有分離，另一開集必是第一個的補集。",
          "en": "In any separation, the second open is exactly the complement of the first."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不交與覆蓋缺一不可，否則不能推出補集等式。",
            "en": "Both disjointness and coverage are required for the complement equality."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S05",
        "uses": [
          "S04"
        ],
        "before_tex": "U,V\\in\\tau,\\quad V=X\\setminus U",
        "after_tex": "U\\text{ is nontrivial clopen}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "分離的兩開集使 U 同時閉，且非空非全。",
          "en": "The two open sides make U closed as well, and nonempty/proper."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04"
          ],
          "explanation": {
            "zh-Hant": "這是從分離到開閉集的單向推論。",
            "en": "This is the implication from a separation to a clopen witness."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S06",
        "uses": [
          "S03",
          "S05"
        ],
        "before_tex": "\\tau\\cap\\mathcal F=\\{\\varnothing,X\\}",
        "after_tex": "\\text{no separation exists}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先前窮盡表排除任何分離所需的非平凡 U。",
          "en": "The exhaustive table rules out every nontrivial U required by a separation."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "沒有以未找到一個抽樣分離冒充全部排除。",
            "en": "Failure to find a sampled separation has not been substituted for exhaustive exclusion."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S07",
        "uses": [
          "S06"
        ],
        "before_tex": "\\text{no separation of }X",
        "after_tex": "X\\text{ is connected}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "套用連通定義；本例其實每個非空開集都含 a，也能排除不交非空對。",
          "en": "Apply the definition; here every nonempty open also contains a, independently excluding disjoint nonempty pairs."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06"
          ],
          "explanation": {
            "zh-Hant": "補充交集檢查與完整開閉集表一致。",
            "en": "The additional intersection check agrees with the full clopen table."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S08",
        "uses": [
          "G1"
        ],
        "before_tex": "|Y|\\le1",
        "after_tex": "Y\\text{ has no two disjoint nonempty parts covering it}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "空集與單點空間沒有分離，依已宣告慣例皆連通。",
          "en": "Empty and singleton spaces have no separation and are connected under the declared convention."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "邊界結論不依非空開集含 a 的特例論法。",
            "en": "The boundary argument does not rely on the special point a."
          }
        },
        "leaf_basis": "sets"
      }
    ],
    "result": {
      "zh-Hant": "本例連通；只有空集與 X 同時開閉。",
      "en": "The example is connected; only empty and X are clopen."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "{a} 與 {b,c} 是開分離。",
          "en": "{a} and {b,c} form an open separation."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "{b,c} 不在拓撲。",
          "en": "{b,c} is not in the topology."
        }
      },
      {
        "claim": {
          "zh-Hant": "有三個點所以不連通。",
          "en": "Three points force disconnectedness."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "連通由拓撲決定，已排除全部分離。",
          "en": "Connectedness is topological; all separations were excluded."
        }
      },
      {
        "claim": {
          "zh-Hant": "圖沒有畫邊，所以此空間不連通。",
          "en": "A drawing without edges makes this space disconnected."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "未指定圖，不能替換拓撲定義。",
          "en": "No graph is specified; its definition cannot replace the 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": "完整有限拓撲上枚舉全部candidate separations/clopen subsets。disconnected需明示witness；connected需排除全部。採空空間connected的vacuous convention，singleton connected。不得混同graph connectivity。",
          "en": "In a complete finite topology, enumerate all candidate separations or clopen subsets. Disconnectedness requires an explicit witness and connectedness requires excluding every one. The empty space is connected under the declared vacuous convention, and every singleton space is connected. Do not conflate this with graph connectivity."
        },
        "source": {
          "mko_id": "mko-connected-topological-space",
          "sha256": "a3956f0120a382824010422c3639f93914810983c5f1dcf47b08e4f3b2d3d96f",
          "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；連通表示沒有兩個非空、不交、聯集為全空間的開集。",
          "en": "Closed sets complement opens; connectedness means no two nonempty disjoint opens have union equal to the whole space."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整有限枚舉；空空間採連通慣例，不使用圖連通。",
      "en": "Complete finite enumeration, with the empty-space connected convention and no graph connectivity."
    },
    "md_symbols": [
      [
        "U,V",
        "分離候選",
        "candidate separation"
      ],
      [
        "\\mathcal F",
        "全部閉集",
        "all closed sets"
      ]
    ],
    "md_error_controls": [
      {
        "title": "排除需窮盡 / Exclusion is exhaustive",
        "zh": "完整開閉集交表涵蓋全部候選。",
        "en": "The full clopen intersection table covers every candidate."
      },
      {
        "title": "空間與圖分離 / Separate spaces from graphs",
        "zh": "不以頂點或邊的視覺印象判斷拓撲連通。",
        "en": "Do not use a visual vertex/edge impression to classify topological connectedness."
      }
    ],
    "md_open_items": []
  },
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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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}