R0052-26852 · q-20261015-advance-20261117-r05-quotient-topology-0077
finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={"mode":"open_image_error","model":{"x_points":[0,1,2,3,4],"x_topology":[[],[0],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[0]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={"mode":"open_image_error","model":{"x_points":[0,1,2,3,4],"x_topology":[[],[0],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[0]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。
操作前
source roles = [quotient-topology-boundary]
操作後
derived = [[mode, subset, source_open, image, saturation, image_open, open_map_claim, certificate]]
條件：無
直接列出此操作及mode的完整來源欄序，不需JavaScript；布林、字串與null角色保持區別，不補欄或以答案形狀猜介面。
規則：source_ordered_fields_without_JS · 依賴：G001
操作前
source roles = [quotient ORIGINAL X, [0, 1, 2, 3, 4], [[], [0], [0, 1, 2, 3, 4]], 否]
操作後
derived = [[0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0]], [[0, 0, 0, 0, 是, 是], [0, 1, 31, 0, 是, 是], [0, 2, 1, 0, 是, 是], [1, 0, 31, 0, 是, 是], [1, 1, 31, 31, 是, 是], [1, 2, 31, 1, 是, 是], [2, 0, 1, 0, 是, 是], [2, 1, 31, 1, 是, 是], [2, 2, 1, 1, 是, 是]]]
條件：無
保留JSON型別，set與open族依來源English序。逐全部open pair驗union及intersection，含空集與全集；有限族的二元閉包給任意union閉包，不由抽樣認定topology。
規則：Topology_full_typed_carrier_and_ALL_pair_axioms · 依賴：G001, S001
操作前
source roles = [[0, 1, 2, 3, 4], [a, b], [a, a, a, b, b]]
操作後
derived = [[[0, a], [1, a], [2, a], [3, b], [4, b]], [[a, [0, 1, 2]], [b, [3, 4]]], 是]
條件：無
依原始X次序綁定q再排序，對每個typed Y點取完整fiber。total map保每個x恰屬一fiber；surjective才保每fiber非空。
規則：Quotient_ORIGINAL_domain_alignment_ALL_fibers · 依賴：G001, S002
操作前
source roles = [[0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0]], [a, b], [[0, a], [1, a], [2, a], [3, b], [4, b]]]
操作後
derived = [[[0, [], [], 是], [1, [a], [0, 1, 2], 否], [2, [b], [3, 4], 否], [3, [a, b], [0, 1, 2, 3, 4], 是]], [[a, b], []]]
條件：無
逐Y的所有subsets，以完整反像在X是否open定義IFF。反像保空、全集、任意union與有限intersection，故生成topology且此IFF逐subset唯一決定它；此唯一性不需要surjection。
規則：Quotient_ALL_target_subsets_PREIMAGE_IFF_unique_final_topology · 依賴：G001, S003
操作前
source roles = [quotient GENERATED final topology, [a, b], [[a, b], []], 否]
操作後
derived = [[a, b], [[a, b], []], [[0, 0, 3, 3, 是, 是], [0, 1, 3, 0, 是, 是], [1, 0, 3, 0, 是, 是], [1, 1, 0, 0, 是, 是]]]
條件：無
保留JSON型別，set與open族依來源English序。逐全部open pair驗union及intersection，含空集與全集；有限族的二元閉包給任意union閉包，不由抽樣認定topology。
規則：Topology_full_typed_carrier_and_ALL_pair_axioms · 依賴：G001, S004
操作前
source roles = [[[], [0, 1, 2, 3, 4], [0]], [[0, a], [1, a], [2, a], [3, b], [4, b]]]
操作後
derived = [[[], [0, 1, 2, 3, 4]], 是]
條件：無
source opens依原canonical序逐一核A=q^-1(q(A))；只有saturated opens才對應完整fiber聯集。surjective時q^-1在Y subsets上單射，與這些opens互逆，不能把所有source opens當商開集。
規則：Quotient_ALL_source_open_saturation_correspondence · 依賴：G001, S005
操作前
source roles = [[0], [[0, a], [1, a], [2, a], [3, b], [4, b]], [[], [0, 1, 2, 3, 4], [0]], [[a, b], []]]
操作後
derived = [[a], [0, 1, 2], 否, 是, 否]
條件：無
先取q(A)，再取其完整反像；若切開某fiber，saturation嚴格擴大。source open不保image open；image的openness須驗saturation，不是A本身。
規則：Quotient_actual_subset_image_FULL_saturation_separate_openness · 依賴：G001, S006
S007 · premise_and_result_audit — 反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。S006 · premise_and_result_audit — 反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。S005 · premise_and_result_audit — 反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。S004 · premise_and_result_audit — 反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。S003 · premise_and_result_audit — 反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。S002 · premise_and_result_audit — 反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。S001 · premise_and_result_audit — 反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。(open_image_error, [0], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)
依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。
覆核日期：2026-10-26
原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: a32d8d630b30118254ac9348f4aafa5e786d2e18add9110f1143644d19be31c7candidate record SHA-256: 86c18ce9d1f41283c1f5f2bd853d0130a713f46ab2e6dd88083db1457b8a9764semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2Lean: NOT_RUN · human review: NOT_PERFORMED