{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-hausdorff-space-01",
  "mko_id": "mko-hausdorff-space",
  "title": {
    "zh-Hant": "三點離散空間的全部點對分離",
    "en": "Separating every pair in a discrete three-point space"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-hausdorff-space-01",
    "source": {
      "mko_id": "mko-hausdorff-space",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "084615a021bcdce41165bfa9dd88865f68a94405aa7108b97c7353e8f40ac5b3",
      "english_sha256": "a145da737e1f922a302fc3731116b71d26eb726e2ce1196d7c1ae73b4f07b46d"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "sets",
      "topology",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "完整核三點離散空間的 Hausdorff 性，並對照一個失敗點對。",
      "en": "Exhaustively audit Hausdorffness of a discrete three-point space and contrast a failed pair."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X={a,b,c}, tau=P(X)；對照拓撲 sigma={空集,{a},{a,b},X}。",
          "en": "X={a,b,c}, tau=P(X); contrast sigma={empty,{a},{a,b},X}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "\\tau=\\{\\varnothing,\\{a\\},\\{b\\},\\{c\\},\\{a,b\\},\\{a,c\\},\\{b,c\\},X\\}",
        "after_tex": "\\tau\\text{ is a topology}",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "八個子集全列，聯交仍是子集。",
          "en": "All eight subsets are listed and unions/intersections remain subsets."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "清單是完整冪集，不是樣本。",
            "en": "The list is the complete power set, not a sample."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S02",
        "uses": [
          "G1"
        ],
        "before_tex": "x\\ne y\\in X",
        "after_tex": "\\{\\{a,b\\},\\{a,c\\},\\{b,c\\}\\}\\text{ are all unordered pairs}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三點有且只有這三個無序異點對。",
          "en": "These are exactly the three unordered pairs of distinct points."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "反向有序對可交換見證，不需重複另一證明。",
            "en": "Reversing an ordered pair swaps its witnesses."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S03",
        "uses": [
          "S01",
          "S02"
        ],
        "before_tex": "(x,y)=(a,b)",
        "after_tex": "U=\\{a\\},\\ V=\\{b\\},\\ U\\cap V=\\varnothing",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩個單點集均開並含正確點。",
          "en": "Both singletons are open and contain the assigned points."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "不僅不交，也逐一核點的包含。",
            "en": "Both disjointness and containment of the assigned points are checked."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S04",
        "uses": [
          "S01",
          "S02"
        ],
        "before_tex": "(x,y)=(a,c)",
        "after_tex": "U=\\{a\\},\\ V=\\{c\\},\\ U\\cap V=\\varnothing",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二點對使用對應的開單點見證。",
          "en": "Use the corresponding open singleton witnesses for the second pair."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "見證都在完整拓撲內。",
            "en": "Both witnesses belong to the complete topology."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S05",
        "uses": [
          "S01",
          "S02"
        ],
        "before_tex": "(x,y)=(b,c)",
        "after_tex": "U=\\{b\\},\\ V=\\{c\\},\\ U\\cap V=\\varnothing",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第三點對亦分離，至此沒有未核異點對。",
          "en": "The third pair is separated, exhausting all distinct pairs."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "全稱成立來自全部點對而非若干成功。",
            "en": "The universal result comes from all pairs, not several successes."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S06",
        "uses": [
          "S02",
          "S03",
          "S04",
          "S05"
        ],
        "before_tex": "\\forall x\\ne y\\ \\exists U,V\\text{ disjoint open neighborhoods}",
        "after_tex": "(X,\\tau)\\text{ is Hausdorff}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "以上三項和交換對稱性完成定義。",
          "en": "The three audits and swapping symmetry establish the definition."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "S03",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "空與單點空間則沒有異點對，條件真。",
            "en": "Empty and singleton spaces have no distinct pairs, so the condition is vacuous."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S07",
        "uses": [
          "G1"
        ],
        "before_tex": "\\sigma=\\{\\varnothing,\\{a\\},\\{a,b\\},X\\}",
        "after_tex": "\\mathcal N_\\sigma(c)=\\{X\\},\\quad\\mathcal N_\\sigma(a)=\\{\\{a\\},\\{a,b\\},X\\}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對照族是含兩端的鏈拓撲；列出 c 與 a 的全部開鄰域。",
          "en": "The contrast family is a chain topology with endpoints; list every open neighborhood of c and a."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "這個失敗點對有完整鄰域枚舉，不是選壞一對見證。",
            "en": "The failed pair has exhaustive neighborhood lists, not merely one badly chosen pair of witnesses."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S08",
        "uses": [
          "S07"
        ],
        "before_tex": "U\\in\\mathcal N_\\sigma(c),\\quad V\\in\\mathcal N_\\sigma(a)",
        "after_tex": "U\\cap V=V\\ne\\varnothing",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "c 只能取 X，故與 a 的任何開鄰域都相交，sigma 非 Hausdorff。",
          "en": "c only has X, so every open neighborhood of a meets it; sigma is not Hausdorff."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S07"
          ],
          "explanation": {
            "zh-Hant": "一個完整失敗點對足以否定全稱性。",
            "en": "One exhaustively failed pair refutes the universal property."
          }
        },
        "leaf_basis": "sets"
      }
    ],
    "result": {
      "zh-Hant": "離散 tau 為 Hausdorff；鏈拓撲 sigma 由 (a,c) 點對證非 Hausdorff。",
      "en": "Discrete tau is Hausdorff; the pair (a,c) proves chain topology sigma is not."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "成功分離一對就證 Hausdorff。",
          "en": "Separating one pair proves Hausdorffness."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "定義需每一異點對。",
          "en": "Every distinct pair is required."
        }
      },
      {
        "claim": {
          "zh-Hant": "只選一組相交鄰域就證非 Hausdorff。",
          "en": "One intersecting choice proves non-Hausdorffness."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "必須排除該點對的所有不交見證；S07 完整列出才足夠。",
          "en": "All disjoint witnesses for that pair must be excluded; the full list at S07 is needed."
        }
      },
      {
        "claim": {
          "zh-Hant": "所有無限 Hausdorff 空間也離散。",
          "en": "Every infinite Hausdorff space is discrete."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "通常實數線 Hausdorff，但單點不開。",
          "en": "The usual real line is Hausdorff but its singletons are not open."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "必須forall x≠y存在open U,V，使x∈U、y∈V且U∩V=empty。一個失敗pair即可證non-Hausdorff；有限sample成功不足以證一般空間。本輪完整有限topology exhaustive驗證。empty與singleton vacuously Hausdorff；finite Hausdorff implies discrete不可外推至infinite spaces。",
          "en": "For every x≠y there must be open U,V containing x,y respectively with U∩V empty. One exhaustively failed pair proves non-Hausdorffness; finitely sampled successes do not certify a general space. This slice exhaustively audits a complete finite topology. Empty and singleton spaces are vacuously Hausdorff. The finite Hausdorff-implies-discrete corollary must not be extrapolated to infinite spaces."
        },
        "source": {
          "mko_id": "mko-hausdorff-space",
          "sha256": "084615a021bcdce41165bfa9dd88865f68a94405aa7108b97c7353e8f40ac5b3",
          "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": "Hausdorff：每對異點各有包含該點的不交開鄰域；點的開鄰域為含該點的開集。",
          "en": "Hausdorffness: every distinct pair has disjoint open neighborhoods containing the assigned points; an open neighborhood is an open set containing the point."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限全枚舉；不把有限 Hausdorff 推離散的結論外推。",
      "en": "Exhaustive finite enumeration; no extrapolation of the finite Hausdorff/discrete corollary."
    },
    "md_symbols": [
      [
        "\\mathcal N(x)",
        "全部開鄰域",
        "all open neighborhoods"
      ],
      [
        "\\tau,\\sigma",
        "兩個指定拓撲",
        "two specified topologies"
      ]
    ],
    "md_error_controls": [
      {
        "title": "點對完整性 / Pair completeness",
        "zh": "列完整無序異點對再交換方向。",
        "en": "List all unordered distinct pairs and swap directions."
      },
      {
        "title": "失敗證書 / Failure certificate",
        "zh": "非 Hausdorff 需一個點對的所有鄰域選擇都失敗。",
        "en": "A non-Hausdorff certificate needs all neighborhood choices for one pair to fail."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "e8794925db0c4d7267f7a0f2001404f5c259749d819b000219306b1c0f6dd12d",
    "original_source_sha256": "e8794925db0c4d7267f7a0f2001404f5c259749d819b000219306b1c0f6dd12d",
    "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": [],
    "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."
  }
}