{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-compact-subset-closed-in-hausdorff-01",
  "mko_id": "mko-compact-subset-closed-in-hausdorff",
  "title": {
    "zh-Hant": "緊緻把分離鄰域縮為有限個",
    "en": "Compactness reduces separating neighborhoods to finitely many"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-compact-subset-closed-in-hausdorff-01",
    "source": {
      "mko_id": "mko-compact-subset-closed-in-hausdorff",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "13e0a8f9917f4a49a9cbe148c344b3f37d319fb16c750de2bb640ec93553d07d",
      "english_sha256": "b4153bb071eb67ca522480652f8411f2eb944d99e54b0bb5043b582af432b244"
    },
    "mode": "theorem_proof",
    "audience_prerequisites": [
      "sets",
      "topology",
      "hausdorff_def",
      "compact_def"
    ],
    "goal": {
      "zh-Hant": "證 Hausdorff 空間中的緊緻子集 K 閉。",
      "en": "Prove a compact subset K of a Hausdorff space is closed."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X 是 Hausdorff，K 以子空間拓撲緊緻。",
          "en": "X is Hausdorff and K is compact in its subspace topology."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "K=\\varnothing",
        "after_tex": "X\\setminus K=X\\text{ is open}",
        "relation": "case_split",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先處理空 K，空集閉直接來自全集開。",
          "en": "Handle empty K first: its complement is the open whole space."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "此分支不需選 K 內的點或非空子覆蓋。",
            "en": "This branch requires no point in K or nonempty subcover."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S02",
        "uses": [
          "G1"
        ],
        "before_tex": "K\\ne\\varnothing,\\quad x\\in X\\setminus K",
        "after_tex": "x\\ne y\\quad\\forall y\\in K",
        "relation": "implies",
        "rule": "sets",
        "conditions": [
          "K\\ne\\varnothing"
        ],
        "explanation": {
          "zh-Hant": "以下固定任意外點 x；因它不在 K，所以不同於每個 y。",
          "en": "Fix an arbitrary outside point x; it differs from every y in K."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "x 的任意性會在最後給整個補集開。",
            "en": "Arbitrariness of x will establish openness of the whole complement."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "hausdorff_def"
        ],
        "before_tex": "x\\ne y\\in K\\cup\\{x\\}",
        "after_tex": "x\\in U_y,\\ y\\in V_y,\\ U_y,V_y\\text{ open},\\ U_y\\cap V_y=\\varnothing",
        "relation": "theorem_application",
        "rule": "hausdorff_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐 y 使用 Hausdorff 分離，兩側鄰域同時保留。",
          "en": "For each y use Hausdorff separation, retaining both neighborhoods."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "hausdorff_def"
          ],
          "explanation": {
            "zh-Hant": "只核一個 y 的鄰域不能覆蓋整個 K。",
            "en": "Neighborhoods for one y alone need not cover K."
          }
        },
        "leaf_basis": "hausdorff_def"
      },
      {
        "id": "S04",
        "uses": [
          "S03"
        ],
        "before_tex": "y\\in K\\cap V_y\\quad(y\\in K)",
        "after_tex": "K\\subseteq\\bigcup_{y\\in K}(K\\cap V_y)",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個 y 被自己的 V_y 覆蓋，且相交 K 為相對開。",
          "en": "Each y lies in its own V_y, and intersection with K is relatively open."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03"
          ],
          "explanation": {
            "zh-Hant": "不是主張所有 U_y 交集已開；那可能是無限交。",
            "en": "No claim is made yet that the intersection of every U_y is open; that could be infinite."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "compact_def"
        ],
        "before_tex": "(K\\cap V_y)_{y\\in K}\\text{ covers }K",
        "after_tex": "K\\subseteq\\bigcup_{j=1}^{m}V_{y_j},\\quad m\\in\\mathbb N",
        "relation": "theorem_application",
        "rule": "compact_def",
        "conditions": [
          "K\\ne\\varnothing"
        ],
        "explanation": {
          "zh-Hant": "緊緻性選有限多個原索引；因 K 非空可以取 m≥1。",
          "en": "Compactness selects finitely many original indices; K nonempty permits m≥1."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "compact_def"
          ],
          "explanation": {
            "zh-Hant": "有限性是下一步開交成立的必要橋梁。",
            "en": "Finiteness is the essential bridge to openness of the next intersection."
          }
        },
        "leaf_basis": "compact_def"
      },
      {
        "id": "S06",
        "uses": [
          "S03",
          "S05"
        ],
        "before_tex": "U=\\bigcap_{j=1}^{m}U_{y_j}",
        "after_tex": "U\\text{ is open and }x\\in U",
        "relation": "implies",
        "rule": "topology",
        "conditions": [
          "m<\\infty"
        ],
        "explanation": {
          "zh-Hant": "有限交保開，每項含 x，故交仍含 x。",
          "en": "Finite intersections of opens are open, and every factor contains x."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "使用有限而非任意交公理。",
            "en": "The finite-intersection axiom, not arbitrary intersection closure, is used."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S07",
        "uses": [
          "S03",
          "S05",
          "S06"
        ],
        "before_tex": "z\\in U\\cap K",
        "after_tex": "z\\in V_{y_j}\\cap U_{y_j}=\\varnothing\\text{ for some }j",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "若有 z 同在 U 與 K，有限子覆蓋選到含 z 的 V；U 又包含於對應 U，矛盾。",
          "en": "A point z in U and K lies in some selected V, while U lies in the corresponding U-neighborhood, a contradiction."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S05",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "每個 z 可用不同 j；不需一個 V 覆蓋整個 K。",
            "en": "The index j may depend on z; no single V is assumed to cover K."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S08",
        "uses": [
          "S01",
          "S02",
          "S06",
          "S07"
        ],
        "before_tex": "\\forall x\\in X\\setminus K\\ \\exists U_x\\text{ open}:x\\in U_x\\subseteq X\\setminus K",
        "after_tex": "X\\setminus K=\\bigcup_{x\\notin K}U_x\\text{ open};\\quad K\\text{ closed}",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "補集是這些開鄰域的聯，所以開；若補集空，空聯也開。",
          "en": "The complement is the union of these open neighborhoods, hence open; an empty complement is the empty open union."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02",
            "S06",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "只得到正向定理，沒有反轉成閉集皆緊緻。",
            "en": "This establishes only the forward theorem, not that every closed set is compact."
          }
        },
        "leaf_basis": "topology"
      }
    ],
    "result": {
      "zh-Hant": "在所列兩項假設下 K 閉，空 K 亦包含。",
      "en": "Under the two stated hypotheses K is closed, including empty K."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "任意拓撲空間的緊緻子集都閉。",
          "en": "Compact subsets of every topological space are closed."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "在只以空集、全集為開集的兩點空間中，單點集緊緻但不閉。",
          "en": "A singleton in the indiscrete two-point space is compact but not closed."
        }
      },
      {
        "claim": {
          "zh-Hant": "所有 U_y 的無限交必開。",
          "en": "The intersection of all U_y is necessarily open."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "拓撲只保有限交開，需先用緊緻性。",
          "en": "Only finite intersections are guaranteed open; compactness must be used first."
        }
      },
      {
        "claim": {
          "zh-Hant": "實數線的所有閉集都緊緻。",
          "en": "Every closed subset of the real line is compact."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "R 閉；覆蓋 (-n,n), n≥1 沒有有限子覆蓋。",
          "en": "R is closed; the cover (-n,n), n≥1, has no finite subcover."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "對每個x∉K及每個y∈K，以Hausdorff性選不交open neighborhoods U_y含x與V_y含y；{V_y}覆蓋K，compactness給有限subcover。相應有限個U_y的交集是與K不交的open neighborhood of x，故X\\K open。empty compact subset亦closed。Hausdorff假設不可刪；converse也不成立。",
          "en": "For each x outside K and each y in K, choose disjoint open U_y containing x and V_y containing y. The V_y cover K, so compactness supplies a finite subcover. The corresponding finite intersection of U_y is an open neighborhood of x disjoint from K; therefore X minus K is open. Empty K is closed too. The Hausdorff hypothesis cannot be removed, and closedness does not conversely imply compactness in general."
        },
        "source": {
          "mko_id": "mko-compact-subset-closed-in-hausdorff",
          "sha256": "13e0a8f9917f4a49a9cbe148c344b3f37d319fb16c750de2bb640ec93553d07d",
          "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": "hausdorff_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "Hausdorff：每對異點各有不交開鄰域包含指定點。",
          "en": "Hausdorffness: every distinct pair has disjoint open neighborhoods containing the assigned points."
        }
      },
      {
        "id": "compact_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "K 緊緻表示每個相對開覆蓋有有限原索引子覆蓋；相對開集為母開集與 K 之交。",
          "en": "K is compact when every relative-open cover has a finite original-index subcover; relative opens are intersections of ambient opens with K."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "給出覆蓋的一般論證；有限例子不取代此證明，且不證逆命題。",
      "en": "A general cover proof, not replaced by finite instances and not proving a converse."
    },
    "md_symbols": [
      [
        "U_y,V_y",
        "分離鄰域",
        "separating neighborhoods"
      ],
      [
        "m",
        "有限子覆蓋大小",
        "finite-subcover size"
      ]
    ],
    "md_error_controls": [
      {
        "title": "有限交橋梁 / Finite-intersection bridge",
        "zh": "先選有限 V 子覆蓋，才交對應 U。",
        "en": "Select finitely many V's before intersecting the corresponding U's."
      },
      {
        "title": "點對量詞 / Pairwise quantifiers",
        "zh": "對固定外點與每個 K 內點分離，最後再放開外點。",
        "en": "Separate a fixed outside point from every point of K, then release the outside-point quantifier."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "original_source_sha256": "70031bac985789aa1c49e84f0a64049275096117bd97bc22051b96b5a3ce3cac",
    "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": [
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    "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."
  }
}