# 第 26555 題：拓撲連通分支

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本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。

R0052-26555 · q-20261015-advance-20261117-r07-topological-connected-component-0030



## 原題與已知條件

subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]],"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-maximality；資料={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]],"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](https://ocme.evemisslab.com/zh-Hant/mko/mko-topological-connected-component/)





## 逐步推導

S001 · implies操作前

source roles = [component-maximality]

操作後

derived = [[basepoint, component, containing_connected_subsets, maximal, closed, 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。

步驟識別與依賴規則：source_ordered_fields_without_JS · 依賴：G001





S002 · implies操作前

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

操作後

derived = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3], [2, 4, 5]], [[0, 0, 0, 0, 是, 是], [0, 1, 63, 0, 是, 是], [0, 2, 11, 0, 是, 是], [0, 3, 52, 0, 是, 是], [1, 0, 63, 0, 是, 是], [1, 1, 63, 63, 是, 是], [1, 2, 63, 11, 是, 是], [1, 3, 63, 52, 是, 是], [2, 0, 11, 0, 是, 是], [2, 1, 63, 11, 是, 是], [2, 2, 11, 11, 是, 是], [2, 3, 63, 0, 是, 是], [3, 0, 52, 0, 是, 是], [3, 1, 63, 52, 是, 是], [3, 2, 63, 0, 是, 是], [3, 3, 52, 52, 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。

步驟識別與依賴規則：Topology_full_typed_carrier_and_ALL_pair_axioms · 依賴：G001, S001





S003 · implies操作前

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

操作後

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], [0], [5]], [0], 否], [34, [1, 5], [[], [1, 5], [1], [5]], [1], 否], [35, [0, 1, 5], [[], [0, 1, 5], [0, 1], [5]], [0, 1], 否], [36, [2, 5], [[], [2, 5]], null, 是], [37, [0, 2, 5], [[], [0, 2, 5], [0], [2, 5]], [0], 否], [38, [1, 2, 5], [[], [1, 2, 5], [1], [2, 5]], [1], 否], [39, [0, 1, 2, 5], [[], [0, 1, 2, 5], [0, 1], [2, 5]], [0, 1], 否], [40, [3, 5], [[], [3, 5], [3], [5]], [3], 否], [41, [0, 3, 5], [[], [0, 3, 5], [0, 3], [5]], [0, 3], 否], [42, [1, 3, 5], [[], [1, 3, 5], [1, 3], [5]], [1, 3], 否], [43, [0, 1, 3, 5], [[], [0, 1, 3, 5], [0, 1, 3], [5]], [0, 1, 3], 否], [44, [2, 3, 5], [[], [2, 3, 5], [2, 5], [3]], [3], 否], [45, [0, 2, 3, 5], [[], [0, 2, 3, 5], [0, 3], [2, 5]], [0, 3], 否], [46, [1, 2, 3, 5], [[], [1, 2, 3, 5], [1, 3], [2, 5]], [1, 3], 否], [47, [0, 1, 2, 3, 5], [[], [0, 1, 2, 3, 5], [0, 1, 3], [2, 5]], [0, 1, 3], 否], [48, [4, 5], [[], [4, 5]], null, 是], [49, [0, 4, 5], [[], [0, 4, 5], [0], [4, 5]], [0], 否], [50, [1, 4, 5], [[], [1, 4, 5], [1], [4, 5]], [1], 否], [51, [0, 1, 4, 5], [[], [0, 1, 4, 5], [0, 1], [4, 5]], [0, 1], 否], [52, [2, 4, 5], [[], [2, 4, 5]], null, 是], [53, [0, 2, 4, 5], [[], [0, 2, 4, 5], [0], [2, 4, 5]], [0], 否], [54, [1, 2, 4, 5], [[], [1, 2, 4, 5], [1], [2, 4, 5]], [1], 否], [55, [0, 1, 2, 4, 5], [[], [0, 1, 2, 4, 5], [0, 1], [2, 4, 5]], [0, 1], 否], [56, [3, 4, 5], [[], [3, 4, 5], [3], [4, 5]], [3], 否], [57, [0, 3, 4, 5], [[], [0, 3, 4, 5], [0, 3], [4, 5]], [0, 3], 否], [58, [1, 3, 4, 5], [[], [1, 3, 4, 5], [1, 3], [4, 5]], [1, 3], 否], [59, [0, 1, 3, 4, 5], [[], [0, 1, 3, 4, 5], [0, 1, 3], [4, 5]], [0, 1, 3], 否], [60, [2, 3, 4, 5], [[], [2, 3, 4, 5], [2, 4, 5], [3]], [3], 否], [61, [0, 2, 3, 4, 5], [[], [0, 2, 3, 4, 5], [0, 3], [2, 4, 5]], [0, 3], 否], [62, [1, 2, 3, 4, 5], [[], [1, 2, 3, 4, 5], [1, 3], [2, 4, 5]], [1, 3], 否], [63, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3], [2, 4, 5]], [0, 1, 3], 否]], [[0], [1], [0, 1], [2], [3], [0, 3], [1, 3], [0, 1, 3], [4], [2, 4], [5], [2, 5], [4, 5], [2, 4, 5]]]

條件：無

每個非空subset都用O∩A生成全部subspace opens，再找proper非空part及其相對補集同開的separation。不存在者才連通；保numeric mask序，singleton自然連通。

步驟識別與依賴規則：Component_ALL_nonempty_masks_induced_topology_no_clopen_separation · 依賴：G001, S002





S004 · implies操作前

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

操作後

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

條件：無

從完整連通族按包含關係取maximal，不取最大基數。兩分支若交非空，其聯集連通使maximal性迫相同；每點singleton被包含所以覆蓋。分支closure仍連通故closed；有限多closed分支使每個補集closed，因此本有限空間亦open。

步驟識別與依賴規則：Component_inclusion_MAXIMAL_partition_finite_clopen · 依賴：G001, S003





S005 · implies操作前

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

操作後

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

條件：無

所有列出連通集共享basepoint；若其聯集有separation，每個含basepoint的連通集都落同側，另一側必空，矛盾。故聯集連通且包含全部這些集合，正是唯一inclusion-maximal分支。

步驟識別與依賴規則：Component_basepoint_common_point_union_AND_maximality · 依賴：G001, S004









## 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, 1, 3], [[0], [0, 1], [0, 3], [0, 1, 3]], 是, 是, unique_inclusion_maximal_connected_subset)

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。

來源雜湊與覆核憑證
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: dc9f091e3798aeb421346b126a217360a7080680f5b0023069c86e2303186a35
- candidate record SHA-256: 4b019d3b61fd45db2e004b65419f5ec0b92944516839d586e23f4d15b91d29c4
- semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
- question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
- Lean: NOT_RUN · human review: NOT_PERFORMED

[同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261117-r07-topological-connected-component-0030.json)