{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-quotient-topology-01",
  "mko_id": "mko-quotient-topology",
  "title": {
    "zh-Hant": "完整逆像條件決定商拓撲",
    "en": "The full preimage condition determines the quotient topology"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-quotient-topology-01",
    "source": {
      "mko_id": "mko-quotient-topology",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "6d39ac6c71e17262644cf9f224ea59cf647428b87cbb6eaf46dc2b3cdd563097",
      "english_sha256": "23927e8ec4e21af1c3c25981e6764a705877f07d698217deb5e780c1f756269e"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "sets",
      "topology",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "由三點到兩點的滿射計算商拓撲與開像失敗。",
      "en": "Compute a quotient topology for a three-to-two-point surjection and show an open-image failure."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X={a,b,c}, tau_X={空集,{a},X}; Y={u,v}; q(a)=q(c)=u,q(b)=v。",
          "en": "X={a,b,c}, tau_X={empty,{a},X}; Y={u,v}; q(a)=q(c)=u,q(b)=v."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "\\varnothing\\subset\\{a\\}\\subset X",
        "after_tex": "\\tau_X\\text{ is a topology}",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "含兩端鏈對聯交封閉，完整母拓撲合法。",
          "en": "The chain with endpoints is union/intersection closed and qualifies the full parent topology."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "未假設所有單點開。",
            "en": "Not every singleton is assumed open."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S02",
        "uses": [
          "G1"
        ],
        "before_tex": "q:X\\to Y",
        "after_tex": "q^{-1}(\\{u\\})=\\{a,c\\},\\quad q^{-1}(\\{v\\})=\\{b\\}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三個來源各有值，兩個 fiber 非空且分割 X，故 q 是 total surjection。",
          "en": "Every source has a value; the two nonempty fibers partition X, proving total surjectivity."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "商慣例所需滿射已另核，並非從連續性取得。",
            "en": "Surjectivity for the quotient convention is checked separately, not inferred from continuity."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S03",
        "uses": [
          "S02"
        ],
        "before_tex": "U\\subseteq Y",
        "after_tex": "U\\in\\{\\varnothing,\\{u\\},\\{v\\},Y\\}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩點載體的全部四個子集都須測逆像。",
          "en": "All four subsets of the two-point carrier must undergo the preimage test."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02"
          ],
          "explanation": {
            "zh-Hant": "此清單完整，沒有以供給樣本取代。",
            "en": "The list is exhaustive, not a supplied sample."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S02",
          "S03"
        ],
        "before_tex": "q^{-1}(\\varnothing),q^{-1}(\\{u\\}),q^{-1}(\\{v\\}),q^{-1}(Y)",
        "after_tex": "\\varnothing,\\{a,c\\},\\{b\\},X",
        "relation": "equivalent",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "中間兩項不開，只有空集與全集的逆像開。",
          "en": "The middle two are not open; only empty and whole have open preimages."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S02",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "完整母拓撲排除 {a,c} 與 {b}。",
            "en": "The full ambient topology excludes {a,c} and {b}."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S05",
        "uses": [
          "S04"
        ],
        "before_tex": "\\tau_q=\\{U\\subseteq Y:q^{-1}(U)\\in\\tau_X\\}",
        "after_tex": "\\tau_q=\\{\\varnothing,Y\\}\\text{ is a topology, and }q\\text{ continuous}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "這個僅含空集與 Y 的兩項開集族對聯、交封閉；其全部開集的逆像已核為開集。",
          "en": "The indiscrete two-member family is union/intersection closed, and all its open preimages are already open."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04"
          ],
          "explanation": {
            "zh-Hant": "連續性是完整 iff 定義的後果，不能反過來僅由連續性唯一決定。",
            "en": "Continuity follows from the full iff definition; continuity alone does not uniquely determine a topology."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S02",
          "S05"
        ],
        "before_tex": "O=\\{a\\}\\in\\tau_X",
        "after_tex": "q(O)=\\{u\\}\\notin\\tau_q,\\quad q^{-1}(q(O))=\\{a,c\\}\\ne O",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "O 未飽和，其開像不開；完整逆像一定是 fiber 聯。",
          "en": "O is not saturated and its image is not open; full preimages are unions of fibers."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "沒有把任意開像拿來替代商開集定義。",
            "en": "Arbitrary open images are not substituted for the quotient-open definition."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S07",
        "uses": [
          "S05"
        ],
        "before_tex": "W\\text{ open in }Z,\\quad(q\\text{ then }h)^{-1}(W)=q^{-1}(h^{-1}(W))",
        "after_tex": "h^{-1}(W)\\in\\tau_q\\iff q^{-1}(h^{-1}(W))\\in\\tau_X",
        "relation": "equivalent",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對任意 h:Y→Z 及每個 Z 開集，商定義把連續性測試逐項等價化。",
          "en": "For any h:Y to Z and each Z-open, the quotient definition makes the continuity tests equivalent."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "S05"
          ],
          "explanation": {
            "zh-Hant": "需要逐 W 的 iff，不是僅 q 連續的單向條件。",
            "en": "The iff for each W is required, not only one-way continuity of q."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S08",
        "uses": [
          "S07",
          "definition_here"
        ],
        "before_tex": "\\forall W\\in\\tau_Z\\text{ the preceding iff holds}",
        "after_tex": "h\\text{ continuous}\\iff h\\circ q\\text{ continuous}",
        "relation": "equivalent",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "量化全部 W 得 final property；同一逆像 iff 亦唯一決定非滿射 total map 的 final topology。",
          "en": "Quantifying every W gives the final property; the same preimage iff uniquely specifies a final topology even for nonsurjective total maps."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "S07",
            "definition_here"
          ],
          "explanation": {
            "zh-Hant": "非滿射不符合本節 quotient map 命名慣例，但不是 final topology 不唯一。",
            "en": "A nonsurjection fails this node's quotient-map naming convention, not uniqueness of its final topology."
          }
        },
        "leaf_basis": "definition_here"
      }
    ],
    "result": {
      "zh-Hant": "商拓撲為 {空集,Y}；{a} 的開像不是商開集。",
      "en": "The quotient topology is {empty,Y}; the image of open {a} is not quotient-open."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "q({a}) 必開。",
          "en": "q({a}) must be open."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "其完整逆像 {a,c} 不開。",
          "en": "Its full preimage {a,c} is not open."
        }
      },
      {
        "claim": {
          "zh-Hant": "只要求 q 連續即唯一決定目標拓撲。",
          "en": "Continuity of q alone uniquely determines the target topology."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "例如離散兩點來源的恆等 q 對多種目標拓撲皆連續。",
          "en": "For example the identity from a discrete two-point source is continuous for several target topologies."
        }
      },
      {
        "claim": {
          "zh-Hant": "非滿射使完整逆像 iff 不能唯一決定 final topology。",
          "en": "Nonsurjectivity prevents the full preimage iff from uniquely defining a final topology."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "該 iff 對每個目標子集直接判開性，未命中點也不破壞唯一性。",
          "en": "The iff directly decides openness of every target subset; unhit points do not destroy uniqueness."
        }
      }
    ],
    "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 models先驗q為total surjection並完整列出fibers。定義tau_q={U⊆Y:q^{-1}(U)∈tau_X}，直接枚舉並驗拓撲公理、q的連續性與final-property：h:Y→Z連續當且僅當h∘q連續。只有saturated open subsets of X能作商開集的完整逆像；直接取open images一般錯誤。對任何total map（含非滿射），上述完整逆像的「當且僅當」條件仍唯一決定Y上的final topology；只有要求q連續則通常不唯一。未被命中的點不妨礙preimage-iff定義，但依此節點採用的慣例，非滿射q不稱為quotient map。",
          "en": "In finite models first qualify q:X→Y as a total surjection and enumerate every fiber. Define tau_q={U⊆Y:q^(-1)(U) is open in X}; enumerate the entire family and verify its topology axioms, continuity of q and the final property: h:Y→Z is continuous if and only if h∘q is continuous. Only saturated open subsets of X can be full preimages of quotient opens; images of arbitrary opens need not be open. For every total map, including a nonsurjection, the defining preimage-if-and-only-if condition still uniquely determines the final topology on Y. Requiring only continuity of q is generally weaker and does not uniquely determine that topology. Unhit points do not invalidate this definition, but under the convention used in this node a nonsurjection is not called a quotient map."
        },
        "source": {
          "mko_id": "mko-quotient-topology",
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          "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": "滿射 total q 的商拓撲以 U 開 iff q^{-1}(U) 開定義；連續性只要求正向開集逆像條件；fiber 是單點逆像。",
          "en": "For total surjective q, a quotient subset U is open iff q^{-1}(U) is open; continuity alone requires only open preimages; a fiber is a singleton preimage."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "本例採 total surjection 的商映射慣例；一般 total map 的 final topology 仍唯一。",
      "en": "Use the total-surjection quotient convention; the final topology of any total map remains uniquely defined."
    },
    "md_symbols": [
      [
        "q",
        "滿射商映射",
        "surjective quotient map"
      ],
      [
        "\\tau_q",
        "逆像定義的商拓撲",
        "preimage-defined quotient topology"
      ]
    ],
    "md_error_controls": [
      {
        "title": "飽和性 / Saturation",
        "zh": "開像不等於商開；先測完整 fiber 聯的逆像。",
        "en": "An open image is not automatically quotient-open; test its full fiber-union preimage."
      },
      {
        "title": "定義強度 / Definition strength",
        "zh": "完整 iff 與只要求連續明確分開。",
        "en": "Keep the full iff separate from continuity alone."
      }
    ],
    "md_open_items": []
  },
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}