{ "schema_version": "ocme-derivation-reader-v1", "guide_id": "derivation-mko-homeomorphism-01", "mko_id": "mko-homeomorphism", "title": { "zh-Hant": "兩個有限鏈狀空間的同胚", "en": "A homeomorphism between finite chain spaces" }, "contract": { "schema_version": "ocme-derivation-guide-v0.1", "guide_id": "derivation-mko-homeomorphism-01", "source": { "mko_id": "mko-homeomorphism", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "sha256": "2869b47d1ae828f6ec6ed61a0f7aa9d72ef812e946cb7b04184b3592d2f384de", "english_sha256": "d5057b40272d1d558ba56008bae40cf826c2405ad5dd379b5d22dd4562bcbc89" }, "mode": "worked_example", "audience_prerequisites": [ "sets", "topology", "definition_here" ], "goal": { "zh-Hant": "用全部逆像核對重命名映射是同胚。", "en": "Audit a relabeling homeomorphism using every open preimage." }, "givens": [ { "id": "G1", "statement": { "zh-Hant": "X={a,b,c}, tau_X={空集,{a},{a,b},X}；Y={u,v,w}, tau_Y={空集,{u},{u,v},Y}。", "en": "X={a,b,c}, tau_X={empty,{a},{a,b},X}; Y={u,v,w}, tau_Y={empty,{u},{u,v},Y}." } }, { "id": "G2", "statement": { "zh-Hant": "h(a)=u,h(b)=v,h(c)=w。", "en": "h(a)=u,h(b)=v,h(c)=w." } } ], "steps": [ { "id": "S01", "uses": [ "G1" ], "before_tex": "\\tau_X,\\tau_Y\\text{ are finite inclusion chains}", "after_tex": "\\tau_X,\\tau_Y\\text{ satisfy all topology axioms}", "relation": "implies", "rule": "topology", "conditions": [], "explanation": { "zh-Hant": "各族含空集全集，任意聯及有限交仍是鏈中的極端項。", "en": "Each contains both endpoints, and arbitrary unions/finite intersections remain extremal chain members." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "兩邊公理各自核對，不能只因點數相同就假定。", "en": "Both topologies are checked separately, not inferred from equal cardinality." } }, "leaf_basis": "topology" }, { "id": "S02", "uses": [ "G1", "G2" ], "before_tex": "h:a\\mapsto u,\\ b\\mapsto v,\\ c\\mapsto w", "after_tex": "h\\text{ is total and bijective}", "relation": "implies", "rule": "sets", "conditions": [], "explanation": { "zh-Hant": "三個來源各有一值，三個像互異且耗盡 Y。", "en": "Each source has one value; the three images are distinct and exhaust Y." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "G2" ], "explanation": { "zh-Hant": "因此逆函數三項也有唯一指定。", "en": "Thus the inverse function also has three uniquely specified entries." } }, "leaf_basis": "sets" }, { "id": "S03", "uses": [ "G2" ], "before_tex": "h^{-1}(\\varnothing),\\ h^{-1}(\\{u\\})", "after_tex": "\\varnothing,\\ \\{a\\}", "relation": "equivalent", "rule": "sets", "conditions": [], "explanation": { "zh-Hant": "逐點讀表得到目標前兩個開集的逆像。", "en": "Read the map table to obtain preimages of the first two target opens." }, "reverse_audit": { "kind": "inverse_equivalence", "required_ids": [ "G2" ], "explanation": { "zh-Hant": "逆像是 X 的子集，與反函數的方向不要混淆。", "en": "These preimages are subsets of X; do not confuse preimage notation with inverse-map direction." } }, "leaf_basis": "sets" }, { "id": "S04", "uses": [ "G1", "G2", "definition_here" ], "before_tex": "h^{-1}(\\{u,v\\}),\\ h^{-1}(Y)", "after_tex": "\\{a,b\\},\\ X", "relation": "implies", "rule": "definition_here", "conditions": [], "explanation": { "zh-Hant": "後兩個逆像也在 tau_X；四個目標開集皆已核完，故 h 連續。", "en": "Both remaining preimages lie in tau_X; all four target opens are now checked, so h is continuous." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "G2", "definition_here" ], "explanation": { "zh-Hant": "連續性不是僅核單點的映射值，而是全部開集逆像。", "en": "Continuity is a check of every open preimage, not merely point values." } }, "leaf_basis": "definition_here" }, { "id": "S05", "uses": [ "S02" ], "before_tex": "k=h^{-1}", "after_tex": "k(u)=a,\\ k(v)=b,\\ k(w)=c", "relation": "definition", "rule": "definition_here", "conditions": [], "explanation": { "zh-Hant": "由雙射逐項建構逆函數，滿足 k h 與 h k 都是恆等。", "en": "Construct the inverse entry by entry; both compositions are identities." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "S02" ], "explanation": { "zh-Hant": "先前雙射前提讓此反函數合法。", "en": "The established bijectivity licenses this inverse function." } }, "leaf_basis": "definition_here" }, { "id": "S06", "uses": [ "G1", "S05", "definition_here" ], "before_tex": "k^{-1}(\\varnothing),k^{-1}(\\{a\\}),k^{-1}(\\{a,b\\}),k^{-1}(X)", "after_tex": "\\varnothing,\\{u\\},\\{u,v\\},Y", "relation": "implies", "rule": "definition_here", "conditions": [], "explanation": { "zh-Hant": "完整核四個來源開集對 k 的逆像，都在 tau_Y，故逆函數連續。", "en": "Check all four X-opens under k; every preimage is in tau_Y, proving inverse continuity." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S05", "definition_here" ], "explanation": { "zh-Hant": "並未由 h 連續與雙射直接跳到逆連續。", "en": "Inverse continuity was not inferred merely from continuity and bijectivity of h." } }, "leaf_basis": "definition_here" }, { "id": "S07", "uses": [ "S02", "S04", "S06" ], "before_tex": "h\\text{ bijective},\\quad h,k\\text{ continuous}", "after_tex": "h\\text{ is a homeomorphism}", "relation": "definition", "rule": "definition_here", "conditions": [], "explanation": { "zh-Hant": "三個責任合在一起才符合定義。", "en": "Only the three established obligations together meet the definition." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "S02", "S04", "S06" ], "explanation": { "zh-Hant": "此結論保存開集結構，不附送任何未指定距離資料。", "en": "The conclusion preserves open-set structure and supplies no unspecified distance data." } }, "leaf_basis": "definition_here" }, { "id": "S08", "uses": [ "G1", "S07" ], "before_tex": "A=\\{a,b\\}\\in\\tau_X", "after_tex": "h(A)=\\{u,v\\}\\in\\tau_Y", "relation": "implies", "rule": "sets", "conditions": [], "explanation": { "zh-Hant": "以一個開集像回讀核對結構；一般開性由逆函數連續已證。", "en": "Read back one open image as a consistency check; general openness follows from the established inverse continuity." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S07" ], "explanation": { "zh-Hant": "這個特例不是 S06 全部逆像核對的替代。", "en": "This instance does not replace the complete preimage audit at S06." } }, "leaf_basis": "sets" } ], "result": { "zh-Hant": "h 是這兩個指定有限拓撲間的同胚。", "en": "h is a homeomorphism between the specified finite topologies." }, "negative_controls": [ { "claim": { "zh-Hant": "同點數空間的每個雙射都是同胚。", "en": "Every bijection between equal-cardinality spaces is a homeomorphism." }, "expected": "reject", "reason": { "zh-Hant": "從具離散拓撲的兩點空間到只以空集、全集為開集的兩點空間，其恆等雙射連續，但逆映射不連續。", "en": "The identity from a discrete two-point space to an indiscrete one is continuous but its inverse is not." } }, { "claim": { "zh-Hant": "h 自動是等距映射。", "en": "h is automatically an isometry." }, "expected": "reject", "reason": { "zh-Hant": "本題未給度量，且同胚只核拓撲。", "en": "No metrics were given; homeomorphism checks topology only." } }, { "claim": { "zh-Hant": "只核 h^{-1}({u}) 就完成連續性。", "en": "Checking only h^{-1}({u}) completes continuity." }, "expected": "reject", "reason": { "zh-Hant": "需核全部目標開集。", "en": "Every target open requires checking." } } ], "status": "candidate", "gaps": [], "verification": { "structure": "checked_local_script", "mathematics": "candidate_argument", "formalization": "not_formalized" }, "rule_glossary": [ { "id": "mko_definition", "kind": "cited", "statement": { "zh-Hant": "有限拓撲題完整列出來源與目標拓撲，逐一核目標開集的逆像及反向責任。有限離散空間任意雙射為同胚只限該範圍；同胚不升格成等距、可微或幾何全等。", "en": "Complete source and target finite topologies permit exact audits of every open-set preimage in both directions. Every bijection between finite discrete spaces is a homeomorphism only within that scope. A homeomorphism alone does not establish isometry, differentiability or geometric congruence." }, "source": { "mko_id": "mko-homeomorphism", "sha256": "2869b47d1ae828f6ec6ed61a0f7aa9d72ef812e946cb7b04184b3592d2f384de", "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": "雙射的逆函數滿足兩個合成皆恆等；同胚是雙射且正反函數都連續；連續以全部開集逆像為開定義。", "en": "A bijection's inverse has both compositions equal to identities; a homeomorphism is a bijection continuous in both directions, and continuity tests all open preimages." } } ], "revision_log": [], "numeric_checks": [] }, "presentation": { "md_scope": { "zh-Hant": "完整有限拓撲；不宣稱等距、可微或幾何全等。", "en": "Complete finite topologies; no isometry, differentiability or congruence claim." }, "md_symbols": [ [ "h", "重命名映射", "relabeling map" ], [ "k", "反函數", "inverse function" ] ], "md_error_controls": [ { "title": "兩個方向 / Both directions", "zh": "分別核 h 與反函數的全部開集逆像。", "en": "Audit every open preimage for h and its inverse separately." }, { "title": "結構不升格 / No structure promotion", "zh": "同胚不等於等距、可微或剛體全等。", "en": "Homeomorphism does not mean isometry, differentiability or rigid congruence." } ], "md_open_items": [] }, "revision": { "id": "ocme-new142-reader-adoption-20261021", "collection": "coverage-20261021", "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "adopted_source_sha256": "569c899605b872432df600bc9e60f4e0a755fcd15015d67a2f0f72cb11dbe8dc", "original_source_sha256": "c108be020f742133584c559dbb70070aa0c7766974d449f9c8a8484a251835dc", "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": [ "BR142-P030" ], "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." } }