第 26851 題：商拓撲

原題與已知條件

finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={"mode":"non_surjective","model":{"x_points":[0,1,2,3],"x_topology":[[],[0],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]}}。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":"non_surjective","model":{"x_points":[0,1,2,3],"x_topology":[[],[0],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

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。

• mko-quotient-topology

逐步推導

source roles = [quotient-topology-boundary]

derived = [[mode, fibers, surjective, final_topology, final_exists, quotient_map_claim, certificate]]

source roles = [quotient ORIGINAL X, [0, 1, 2, 3], [[], [0], [0, 1, 2, 3]], 否]

derived = [[0, 1, 2, 3], [[], [0, 1, 2, 3], [0]], [[0, 0, 0, 0, 是, 是], [0, 1, 15, 0, 是, 是], [0, 2, 1, 0, 是, 是], [1, 0, 15, 0, 是, 是], [1, 1, 15, 15, 是, 是], [1, 2, 15, 1, 是, 是], [2, 0, 1, 0, 是, 是], [2, 1, 15, 1, 是, 是], [2, 2, 1, 1, 是, 是]]]

source roles = [[0, 1, 2, 3], [y0, y1, y2, y3, y4], [y0, y1, y2, y3]]

derived = [[[0, y0], [1, y1], [2, y2], [3, y3]], [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否]

source roles = [[0, 1, 2, 3], [[], [0, 1, 2, 3], [0]], [y0, y1, y2, y3, y4], [[0, y0], [1, y1], [2, y2], [3, y3]]]

derived = [[[0, [], [], 是], [1, [y0], [0], 是], [2, [y1], [1], 否], [3, [y0, y1], [0, 1], 否], [4, [y2], [2], 否], [5, [y0, y2], [0, 2], 否], [6, [y1, y2], [1, 2], 否], [7, [y0, y1, y2], [0, 1, 2], 否], [8, [y3], [3], 否], [9, [y0, y3], [0, 3], 否], [10, [y1, y3], [1, 3], 否], [11, [y0, y1, y3], [0, 1, 3], 否], [12, [y2, y3], [2, 3], 否], [13, [y0, y2, y3], [0, 2, 3], 否], [14, [y1, y2, y3], [1, 2, 3], 否], [15, [y0, y1, y2, y3], [0, 1, 2, 3], 是], [16, [y4], [], 是], [17, [y0, y4], [0], 是], [18, [y1, y4], [1], 否], [19, [y0, y1, y4], [0, 1], 否], [20, [y2, y4], [2], 否], [21, [y0, y2, y4], [0, 2], 否], [22, [y1, y2, y4], [1, 2], 否], [23, [y0, y1, y2, y4], [0, 1, 2], 否], [24, [y3, y4], [3], 否], [25, [y0, y3, y4], [0, 3], 否], [26, [y1, y3, y4], [1, 3], 否], [27, [y0, y1, y3, y4], [0, 1, 3], 否], [28, [y2, y3, y4], [2, 3], 否], [29, [y0, y2, y3, y4], [0, 2, 3], 否], [30, [y1, y2, y3, y4], [1, 2, 3], 否], [31, [y0, y1, y2, y3, y4], [0, 1, 2, 3], 是]], [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []]]

source roles = [quotient GENERATED final topology, [y0, y1, y2, y3, y4], [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []], 否]

derived = [[y0, y1, y2, y3, y4], [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []], [[0, 0, 31, 31, 是, 是], [0, 1, 31, 15, 是, 是], [0, 2, 31, 17, 是, 是], [0, 3, 31, 1, 是, 是], [0, 4, 31, 16, 是, 是], [0, 5, 31, 0, 是, 是], [1, 0, 31, 15, 是, 是], [1, 1, 15, 15, 是, 是], [1, 2, 31, 1, 是, 是], [1, 3, 15, 1, 是, 是], [1, 4, 31, 0, 是, 是], [1, 5, 15, 0, 是, 是], [2, 0, 31, 17, 是, 是], [2, 1, 31, 1, 是, 是], [2, 2, 17, 17, 是, 是], [2, 3, 17, 1, 是, 是], [2, 4, 17, 16, 是, 是], [2, 5, 17, 0, 是, 是], [3, 0, 31, 1, 是, 是], [3, 1, 15, 1, 是, 是], [3, 2, 17, 1, 是, 是], [3, 3, 1, 1, 是, 是], [3, 4, 17, 0, 是, 是], [3, 5, 1, 0, 是, 是], [4, 0, 31, 16, 是, 是], [4, 1, 31, 0, 是, 是], [4, 2, 17, 16, 是, 是], [4, 3, 17, 0, 是, 是], [4, 4, 16, 16, 是, 是], [4, 5, 16, 0, 是, 是], [5, 0, 31, 0, 是, 是], [5, 1, 15, 0, 是, 是], [5, 2, 17, 0, 是, 是], [5, 3, 1, 0, 是, 是], [5, 4, 16, 0, 是, 是], [5, 5, 0, 0, 是, 是]]]

source roles = [[[], [0, 1, 2, 3], [0]], [[0, y0], [1, y1], [2, y2], [3, y3]]]

derived = [[[], [0, 1, 2, 3], [0]], 否]

source roles = [[[0, y0], [1, y1], [2, y2], [3, y3]], [y0, y1, y2, y3, y4], [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []]]

derived = [是, 否]

逐步反向核對

• 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。

結論

(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)

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。

本題覆核與來源綁定

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。