{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-continuous-image-of-compact-01",
  "mko_id": "mko-continuous-image-of-compact",
  "title": {
    "zh-Hant": "拉回覆蓋再推回索引",
    "en": "Pull back covers and return the same indices"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-continuous-image-of-compact-01",
    "source": {
      "mko_id": "mko-continuous-image-of-compact",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "033a72b2e9f25c030d4348c2fe05cda62d3477182efb9d2fb0e32b8c1178138c",
      "english_sha256": "86415941d2cdb6419d41013c6d3c1611923949c466f9e07156b1aa77671625b5"
    },
    "mode": "theorem_proof",
    "audience_prerequisites": [
      "sets",
      "topology",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "以明示有限映射呈現緊緻像的覆蓋論證。",
      "en": "Demonstrate the compact-image cover proof with an explicit finite map."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X={a,b}, tau_X={空集,{a},{b},X}; Y={u,v}, tau_Y={空集,{u},Y}; K=X。",
          "en": "X={a,b}, tau_X={empty,{a},{b},X}; Y={u,v}, tau_Y={empty,{u},Y}; K=X."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "f(a)=f(b)=u。",
          "en": "f(a)=f(b)=u."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "\\tau_X=\\mathcal P(X),\\quad\\varnothing\\subset\\{u\\}\\subset Y",
        "after_tex": "\\tau_X,\\tau_Y\\text{ are topologies}",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "來源是全部子集；目標是含兩端的有限鏈，兩邊公理均成立。",
          "en": "The source is the power set and the target a finite chain with endpoints; both satisfy every axiom."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "沒有省略任一目標開集。",
            "en": "No target open has been omitted."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "G2",
          "S01",
          "definition_here"
        ],
        "before_tex": "f^{-1}(\\varnothing),f^{-1}(\\{u\\}),f^{-1}(Y)",
        "after_tex": "\\varnothing,X,X\\in\\tau_X",
        "relation": "implies",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "核全部目標開集的逆像，所以 f 連續。",
          "en": "Check preimages of all target opens, proving continuity of f."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S01",
            "definition_here"
          ],
          "explanation": {
            "zh-Hant": "不需要 f 單射或滿射，實際上兩者皆不成立。",
            "en": "Neither injectivity nor surjectivity is required, and neither holds here."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "definition_here"
        ],
        "before_tex": "K=X=\\{a,b\\}",
        "after_tex": "K\\text{ is compact}",
        "relation": "implies",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對任意覆蓋分別選含 a 與含 b 的原成員，至多兩個便覆蓋 K。",
          "en": "From any cover choose one original member for a and one for b; at most two cover K."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "definition_here"
          ],
          "explanation": {
            "zh-Hant": "有限選取對所有覆蓋適用，不是單一覆蓋成功。",
            "en": "The finite selection applies to every cover, not just one successful instance."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S04",
        "uses": [
          "G2",
          "S02"
        ],
        "before_tex": "f(K)=\\{u\\}\\subseteq\\bigcup_{i\\in I}W_i,\\quad W_i\\in\\tau_Y",
        "after_tex": "K\\subseteq\\bigcup_{i\\in I}(K\\cap f^{-1}(W_i))",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任取像的目標開覆蓋；每個 x∈K 的像被某個 W_i 覆蓋，所以逆像覆蓋 K。",
          "en": "Take an arbitrary ambient-open cover of the image; each x in K has its image in some W_i, so the preimages cover K."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G2",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "只用像點屬性，未要求 f 對全部 Y 滿射。",
            "en": "Only image membership is used, not surjectivity onto Y."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S05",
        "uses": [
          "S02",
          "S04",
          "definition_here"
        ],
        "before_tex": "W_i\\in\\tau_Y",
        "after_tex": "K\\cap f^{-1}(W_i)\\text{ is relatively open in }K",
        "relation": "implies",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "連續性給母逆像開，相交 K 就相對開。",
          "en": "Continuity gives an ambient-open preimage; intersection with K is relatively open."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "S04",
            "definition_here"
          ],
          "explanation": {
            "zh-Hant": "開性前提不能由覆蓋性代替。",
            "en": "Coverage cannot substitute for the openness premise."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S06",
        "uses": [
          "S03",
          "S04",
          "S05",
          "definition_here"
        ],
        "before_tex": "K\\subseteq\\bigcup_{i\\in I}(K\\cap f^{-1}(W_i))",
        "after_tex": "K\\subseteq\\bigcup_{i\\in J}f^{-1}(W_i),\\quad J\\subseteq I\\text{ finite}",
        "relation": "theorem_application",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "用 K 的緊緻性取得有限的原索引子集 J。",
          "en": "Compactness of K supplies a finite subset J of original indices."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S04",
            "S05",
            "definition_here"
          ],
          "explanation": {
            "zh-Hant": "J 的有限性來自已證緊緻性，不是預設 I 有限。",
            "en": "Finiteness of J comes from compactness, not an assumption that I is finite."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S07",
        "uses": [
          "S06"
        ],
        "before_tex": "z\\in f(K),\\ z=f(x),\\ x\\in K",
        "after_tex": "z\\in\\bigcup_{i\\in J}W_i",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "x 被某個選定逆像覆蓋，所以同一 i 的 W_i 包含 z；索引原樣推回。",
          "en": "Some selected preimage contains x, so the same W_i contains z; return exactly the same indices."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06"
          ],
          "explanation": {
            "zh-Hant": "沒有用 f(逆像)=W_i 的錯誤等式；只需包含關係。",
            "en": "No false equality between the image of a preimage and W_i is used; inclusion suffices."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S08",
        "uses": [
          "S04",
          "S07",
          "definition_here"
        ],
        "before_tex": "\\forall(W_i)\\text{ covering }f(K),\\ \\exists J\\text{ finite covering }f(K)",
        "after_tex": "f(K)=\\{u\\}\\text{ is compact but not closed in }Y",
        "relation": "implies",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "前半完成緊緻定義；後半因補集 {v} 不開而不閉，提醒未給 Hausdorff 不能宣稱閉像。",
          "en": "The first part proves compactness; {u} is not closed because {v} is not open, protecting the missing-Hausdorff boundary."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04",
            "S07",
            "definition_here"
          ],
          "explanation": {
            "zh-Hant": "緊緻與閉性由不同責任判斷。",
            "en": "Compactness and closedness are determined by separate obligations."
          }
        },
        "leaf_basis": "definition_here"
      }
    ],
    "result": {
      "zh-Hant": "連續像 {u} 緊緻但在本非 Hausdorff 目標不閉。",
      "en": "The continuous image {u} is compact but not closed in this non-Hausdorff target."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "定理需要 f 單射。",
          "en": "The theorem requires f injective."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本例兩點同像仍已證像緊緻。",
          "en": "Both points have the same image here and compactness still follows."
        }
      },
      {
        "claim": {
          "zh-Hant": "緊緻像推出原空間緊緻。",
          "en": "A compact image forces a compact domain."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "常值映射 (0,1)→{u} 的像緊緻；開覆蓋 (1/n,1)（n≥2）證來源不緊緻。",
          "en": "The constant map (0,1) to {u} has compact image; the cover (1/n,1), n≥2, has no finite subcover of the domain."
        }
      },
      {
        "claim": {
          "zh-Hant": "所有連續緊緻像都閉。",
          "en": "Every continuous compact image is closed."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本例 {u} 的補集 {v} 不開。",
          "en": "The complement {v} of {u} is not open here."
        }
      }
    ],
    "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f(K)的ambient-open cover取逆像，continuity使逆像open且覆蓋K；K的finite subcover再推回覆蓋f(K)。不需要injective或surjective。compact image不反推domain compact；compact preimage或image closed也各需額外假設（後者常需Hausdorff）。",
          "en": "Pull an ambient-open cover of f(K) back along f. Continuity makes preimages open and they cover K. A finite subcover of K pushes forward using the same cover indices to cover f(K). Injectivity and surjectivity are not required. Compactness of an image does not imply compactness of the domain. Compact preimages and closed images require additional hypotheses, often Hausdorff for closedness."
        },
        "source": {
          "mko_id": "mko-continuous-image-of-compact",
          "sha256": "033a72b2e9f25c030d4348c2fe05cda62d3477182efb9d2fb0e32b8c1178138c",
          "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": "連續性要求每個目標開集的逆像開；相對開集為母開集與子空間之交；緊緻表示每個相對開覆蓋有有限原索引子覆蓋；閉集有開補集。",
          "en": "Continuity requires every target-open preimage to be open; relative opens intersect ambient opens with the subspace; compactness requires a finite original-index subcover of every relative-open cover; closed sets have open complements."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整有限拓撲，同時解釋任意目標覆蓋的拉回；不推反命題。",
      "en": "Complete finite topologies with an arbitrary-cover pullback argument; no converse."
    },
    "md_symbols": [
      [
        "W_i",
        "像的目標開覆蓋",
        "target-open cover of the image"
      ],
      [
        "J",
        "保留的原索引",
        "retained original indices"
      ]
    ],
    "md_error_controls": [
      {
        "title": "逆像而非開像 / Preimages, not open images",
        "zh": "用連續性的逆像責任拉回覆蓋。",
        "en": "Pull the cover back by continuity's preimage condition."
      },
      {
        "title": "索引不變 / Preserve indices",
        "zh": "J 從目標原覆蓋取得且原樣推回。",
        "en": "J is taken from and returned to the original target cover."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "d6eaece8c24bc185ce8421028d38a700da364006b5f47dd2476ede583c0cbbbc",
    "original_source_sha256": "d6eaece8c24bc185ce8421028d38a700da364006b5f47dd2476ede583c0cbbbc",
    "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."
  }
}