第 26546 題：拓撲連通分支

原題與已知條件

subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]],"basepoint":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]],"basepoint":0}。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-topological-connected-component

逐步推導

source roles = [component-connected-subsets]

derived = [[basepoint, containing_connected_subsets, combined, connected, certificate]]

source roles = [component ORIGINAL space, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5], [2, 4]], 否]

derived = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5], [2, 4]], [[0, 0, 0, 0, 是, 是], [0, 1, 63, 0, 是, 是], [0, 2, 43, 0, 是, 是], [0, 3, 20, 0, 是, 是], [1, 0, 63, 0, 是, 是], [1, 1, 63, 63, 是, 是], [1, 2, 63, 43, 是, 是], [1, 3, 63, 20, 是, 是], [2, 0, 43, 0, 是, 是], [2, 1, 63, 43, 是, 是], [2, 2, 43, 43, 是, 是], [2, 3, 63, 0, 是, 是], [3, 0, 20, 0, 是, 是], [3, 1, 63, 20, 是, 是], [3, 2, 63, 0, 是, 是], [3, 3, 20, 20, 是, 是]]]

source roles = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5], [2, 4]]]

derived = [[[1, [0], [[], [0]], null, 是], [2, [1], [[], [1]], null, 是], [3, [0, 1], [[], [0, 1]], null, 是], [4, [2], [[], [2]], null, 是], [5, [0, 2], [[], [0, 2], [0], [2]], [0], 否], [6, [1, 2], [[], [1, 2], [1], [2]], [1], 否], [7, [0, 1, 2], [[], [0, 1, 2], [0, 1], [2]], [0, 1], 否], [8, [3], [[], [3]], null, 是], [9, [0, 3], [[], [0, 3]], null, 是], [10, [1, 3], [[], [1, 3]], null, 是], [11, [0, 1, 3], [[], [0, 1, 3]], null, 是], [12, [2, 3], [[], [2, 3], [2], [3]], [2], 否], [13, [0, 2, 3], [[], [0, 2, 3], [0, 3], [2]], [2], 否], [14, [1, 2, 3], [[], [1, 2, 3], [1, 3], [2]], [2], 否], [15, [0, 1, 2, 3], [[], [0, 1, 2, 3], [0, 1, 3], [2]], [2], 否], [16, [4], [[], [4]], null, 是], [17, [0, 4], [[], [0, 4], [0], [4]], [0], 否], [18, [1, 4], [[], [1, 4], [1], [4]], [1], 否], [19, [0, 1, 4], [[], [0, 1, 4], [0, 1], [4]], [0, 1], 否], [20, [2, 4], [[], [2, 4]], null, 是], [21, [0, 2, 4], [[], [0, 2, 4], [0], [2, 4]], [0], 否], [22, [1, 2, 4], [[], [1, 2, 4], [1], [2, 4]], [1], 否], [23, [0, 1, 2, 4], [[], [0, 1, 2, 4], [0, 1], [2, 4]], [0, 1], 否], [24, [3, 4], [[], [3, 4], [3], [4]], [3], 否], [25, [0, 3, 4], [[], [0, 3, 4], [0, 3], [4]], [0, 3], 否], [26, [1, 3, 4], [[], [1, 3, 4], [1, 3], [4]], [1, 3], 否], [27, [0, 1, 3, 4], [[], [0, 1, 3, 4], [0, 1, 3], [4]], [0, 1, 3], 否], [28, [2, 3, 4], [[], [2, 3, 4], [2, 4], [3]], [3], 否], [29, [0, 2, 3, 4], [[], [0, 2, 3, 4], [0, 3], [2, 4]], [0, 3], 否], [30, [1, 2, 3, 4], [[], [1, 2, 3, 4], [1, 3], [2, 4]], [1, 3], 否], [31, [0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0, 1, 3], [2, 4]], [0, 1, 3], 否], [32, [5], [[], [5]], null, 是], [33, [0, 5], [[], [0, 5]], null, 是], [34, [1, 5], [[], [1, 5]], null, 是], [35, [0, 1, 5], [[], [0, 1, 5]], null, 是], [36, [2, 5], [[], [2, 5], [2], [5]], [2], 否], [37, [0, 2, 5], [[], [0, 2, 5], [0, 5], [2]], [2], 否], [38, [1, 2, 5], [[], [1, 2, 5], [1, 5], [2]], [2], 否], [39, [0, 1, 2, 5], [[], [0, 1, 2, 5], [0, 1, 5], [2]], [2], 否], [40, [3, 5], [[], [3, 5]], null, 是], [41, [0, 3, 5], [[], [0, 3, 5]], null, 是], [42, [1, 3, 5], [[], [1, 3, 5]], null, 是], [43, [0, 1, 3, 5], [[], [0, 1, 3, 5]], null, 是], [44, [2, 3, 5], [[], [2, 3, 5], [2], [3, 5]], [2], 否], [45, [0, 2, 3, 5], [[], [0, 2, 3, 5], [0, 3, 5], [2]], [2], 否], [46, [1, 2, 3, 5], [[], [1, 2, 3, 5], [1, 3, 5], [2]], [2], 否], [47, [0, 1, 2, 3, 5], [[], [0, 1, 2, 3, 5], [0, 1, 3, 5], [2]], [2], 否], [48, [4, 5], [[], [4, 5], [4], [5]], [4], 否], [49, [0, 4, 5], [[], [0, 4, 5], [0, 5], [4]], [4], 否], [50, [1, 4, 5], [[], [1, 4, 5], [1, 5], [4]], [4], 否], [51, [0, 1, 4, 5], [[], [0, 1, 4, 5], [0, 1, 5], [4]], [4], 否], [52, [2, 4, 5], [[], [2, 4, 5], [2, 4], [5]], [2, 4], 否], [53, [0, 2, 4, 5], [[], [0, 2, 4, 5], [0, 5], [2, 4]], [2, 4], 否], [54, [1, 2, 4, 5], [[], [1, 2, 4, 5], [1, 5], [2, 4]], [2, 4], 否], [55, [0, 1, 2, 4, 5], [[], [0, 1, 2, 4, 5], [0, 1, 5], [2, 4]], [2, 4], 否], [56, [3, 4, 5], [[], [3, 4, 5], [3, 5], [4]], [4], 否], [57, [0, 3, 4, 5], [[], [0, 3, 4, 5], [0, 3, 5], [4]], [4], 否], [58, [1, 3, 4, 5], [[], [1, 3, 4, 5], [1, 3, 5], [4]], [4], 否], [59, [0, 1, 3, 4, 5], [[], [0, 1, 3, 4, 5], [0, 1, 3, 5], [4]], [4], 否], [60, [2, 3, 4, 5], [[], [2, 3, 4, 5], [2, 4], [3, 5]], [2, 4], 否], [61, [0, 2, 3, 4, 5], [[], [0, 2, 3, 4, 5], [0, 3, 5], [2, 4]], [2, 4], 否], [62, [1, 2, 3, 4, 5], [[], [1, 2, 3, 4, 5], [1, 3, 5], [2, 4]], [2, 4], 否], [63, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5], [2, 4]], [2, 4], 否]], [[0], [1], [0, 1], [2], [3], [0, 3], [1, 3], [0, 1, 3], [4], [2, 4], [5], [0, 5], [1, 5], [0, 1, 5], [3, 5], [0, 3, 5], [1, 3, 5], [0, 1, 3, 5]]]

source roles = [[[0], [1], [0, 1], [2], [3], [0, 3], [1, 3], [0, 1, 3], [4], [2, 4], [5], [0, 5], [1, 5], [0, 1, 5], [3, 5], [0, 3, 5], [1, 3, 5], [0, 1, 3, 5]]]

derived = [[[0, 1, 3, 5], [2, 4]], 是, 是, [是, 是], [是, 是]]

source roles = [0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]]]

derived = [[0, 1, 3, 5], [0, 1, 3, 5], 是, 是]

逐步反向核對

• 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

(0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], [0, 1, 3, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)

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。