R0052-26512 · q-20261015-advance-20261117-r07-topological-connected-component-0087
subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={"mode":"cardinality_not_definition","model":{"points":[0,1,2,3,4],"topology":[[],[0,1,2,3,4],[0],[1,2,3,4]],"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-boundary；資料={"mode":"cardinality_not_definition","model":{"points":[0,1,2,3,4],"topology":[[],[0,1,2,3,4],[0],[1,2,3,4]],"basepoint":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 = [component-boundary]
操作後
derived = [[mode, basepoint, own_component, larger_component, own_size, larger_size, global_maximum_definition, 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 = [component ORIGINAL space, [0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0], [1, 2, 3, 4]], 否]
操作後
derived = [[0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0], [1, 2, 3, 4]], [[0, 0, 0, 0, 是, 是], [0, 1, 31, 0, 是, 是], [0, 2, 1, 0, 是, 是], [0, 3, 30, 0, 是, 是], [1, 0, 31, 0, 是, 是], [1, 1, 31, 31, 是, 是], [1, 2, 31, 1, 是, 是], [1, 3, 31, 30, 是, 是], [2, 0, 1, 0, 是, 是], [2, 1, 31, 1, 是, 是], [2, 2, 1, 1, 是, 是], [2, 3, 31, 0, 是, 是], [3, 0, 30, 0, 是, 是], [3, 1, 31, 30, 是, 是], [3, 2, 31, 0, 是, 是], [3, 3, 30, 30, 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], [[], [0, 1, 2, 3, 4], [0], [1, 2, 3, 4]]]
操作後
derived = [[[1, [0], [[], [0]], null, 是], [2, [1], [[], [1]], null, 是], [3, [0, 1], [[], [0, 1], [0], [1]], [0], 否], [4, [2], [[], [2]], null, 是], [5, [0, 2], [[], [0, 2], [0], [2]], [0], 否], [6, [1, 2], [[], [1, 2]], null, 是], [7, [0, 1, 2], [[], [0, 1, 2], [0], [1, 2]], [0], 否], [8, [3], [[], [3]], null, 是], [9, [0, 3], [[], [0, 3], [0], [3]], [0], 否], [10, [1, 3], [[], [1, 3]], null, 是], [11, [0, 1, 3], [[], [0, 1, 3], [0], [1, 3]], [0], 否], [12, [2, 3], [[], [2, 3]], null, 是], [13, [0, 2, 3], [[], [0, 2, 3], [0], [2, 3]], [0], 否], [14, [1, 2, 3], [[], [1, 2, 3]], null, 是], [15, [0, 1, 2, 3], [[], [0, 1, 2, 3], [0], [1, 2, 3]], [0], 否], [16, [4], [[], [4]], null, 是], [17, [0, 4], [[], [0, 4], [0], [4]], [0], 否], [18, [1, 4], [[], [1, 4]], null, 是], [19, [0, 1, 4], [[], [0, 1, 4], [0], [1, 4]], [0], 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]], null, 是], [23, [0, 1, 2, 4], [[], [0, 1, 2, 4], [0], [1, 2, 4]], [0], 否], [24, [3, 4], [[], [3, 4]], null, 是], [25, [0, 3, 4], [[], [0, 3, 4], [0], [3, 4]], [0], 否], [26, [1, 3, 4], [[], [1, 3, 4]], null, 是], [27, [0, 1, 3, 4], [[], [0, 1, 3, 4], [0], [1, 3, 4]], [0], 否], [28, [2, 3, 4], [[], [2, 3, 4]], null, 是], [29, [0, 2, 3, 4], [[], [0, 2, 3, 4], [0], [2, 3, 4]], [0], 否], [30, [1, 2, 3, 4], [[], [1, 2, 3, 4]], null, 是], [31, [0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0], [1, 2, 3, 4]], [0], 否]], [[0], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3], [4], [1, 4], [2, 4], [1, 2, 4], [3, 4], [1, 3, 4], [2, 3, 4], [1, 2, 3, 4]]]
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。
規則：Component_ALL_nonempty_masks_induced_topology_no_clopen_separation · 依賴：G001, S002
操作前
source roles = [[[0], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3], [4], [1, 4], [2, 4], [1, 2, 4], [3, 4], [1, 3, 4], [2, 3, 4], [1, 2, 3, 4]]]
操作後
derived = [[[0], [1, 2, 3, 4]], 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。
規則：Component_inclusion_MAXIMAL_partition_finite_clopen · 依賴：G001, S003
操作前
source roles = [0, [[0]]]
操作後
derived = [[0], [0], 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。
規則：Component_basepoint_common_point_union_AND_maximality · 依賴：G001, S004
操作前
source roles = [0, [0], [[0], [1, 2, 3, 4]]]
操作後
derived = [[1, 2, 3, 4], 1, 4, 否]
條件：無
依來源canonical序取首較大other分支；本分支不能靠連通擴張，仍可比別分支小，所以maximal不是全域最大基數。
規則：Component_FIRST_canonical_larger_OTHER_not_global_maximum · 依賴：G001, S005
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。(cardinality_not_definition, 0, [0], [1, 2, 3, 4], 1, 4, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)
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: fd72b8ec419c1107789faa92e288f07354ee1540509b5a8211740867a9c1264ccandidate record SHA-256: 877943bfc1b9bdb7f8285cb692e9770e2527174f48ffd93c67f0fecec800d440semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2Lean: NOT_RUN · human review: NOT_PERFORMED