R0043-22030 · q-20261014-advance-20261110-r03-continuous-image-of-compact-0055
compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,1,0],"subset":[0,1,2],"cover":[[0],[1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,1,0],"subset":[0,1,2],"cover":[[0],[1]]}。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 operation=compactimg-cover-pullback;original snapshot and mathematics retained
操作後
explicit ordered source output fields=[image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate]
條件：無
原題歷史介面標記保留，直接列來源有序欄位，不需JavaScript；MVT由原mode選欄序，不補空欄或看答案形狀猜。tag與固定No保原聲稱範圍。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=0,U=[]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1], []], [3, [0, 1, 2], []], [4, [0, 2], []], [5, [1], []], [6, [1, 2], []], [7, [2], []]];empty/full opens present;every result in SAME SOURCE topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S001
操作前
SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=1,U=[0]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1], [0]], [3, [0, 1, 2], [0]], [4, [0, 2], [0]], [5, [0, 1], []], [6, [0, 1, 2], []], [7, [0, 2], []]];empty/full opens present;every result in SAME SOURCE topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S002
操作前
SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=2,U=[0, 1]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [0, 1], []], [1, [0, 1], [0]], [2, [0, 1], [0, 1]], [3, [0, 1, 2], [0, 1]], [4, [0, 1, 2], [0]], [5, [0, 1], [1]], [6, [0, 1, 2], [1]], [7, [0, 1, 2], []]];empty/full opens present;every result in SAME SOURCE topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S003
操作前
SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=3,U=[0, 1, 2]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2], []], [1, [0, 1, 2], [0]], [2, [0, 1, 2], [0, 1]], [3, [0, 1, 2], [0, 1, 2]], [4, [0, 1, 2], [0, 2]], [5, [0, 1, 2], [1]], [6, [0, 1, 2], [1, 2]], [7, [0, 1, 2], [2]]];empty/full opens present;every result in SAME SOURCE topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S004
操作前
SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=4,U=[0, 2]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [0, 2], []], [1, [0, 2], [0]], [2, [0, 1, 2], [0]], [3, [0, 1, 2], [0, 2]], [4, [0, 2], [0, 2]], [5, [0, 1, 2], []], [6, [0, 1, 2], [2]], [7, [0, 2], [2]]];empty/full opens present;every result in SAME SOURCE topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S005
操作前
SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=5,U=[1]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [1], []], [1, [0, 1], []], [2, [0, 1], [1]], [3, [0, 1, 2], [1]], [4, [0, 1, 2], []], [5, [1], [1]], [6, [1, 2], [1]], [7, [1, 2], []]];empty/full opens present;every result in SAME SOURCE topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S006
操作前
SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=6,U=[1, 2]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [1, 2], []], [1, [0, 1, 2], []], [2, [0, 1, 2], [1]], [3, [0, 1, 2], [1, 2]], [4, [0, 1, 2], [2]], [5, [1, 2], [1]], [6, [1, 2], [1, 2]], [7, [1, 2], [2]]];empty/full opens present;every result in SAME SOURCE topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S007
操作前
SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=7,U=[2]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [2], []], [1, [0, 2], []], [2, [0, 1, 2], []], [3, [0, 1, 2], [2]], [4, [0, 2], [2]], [5, [1, 2], []], [6, [1, 2], [2]], [7, [2], [2]]];empty/full opens present;every result in SAME SOURCE topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S008
操作前
TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=0,U=[]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1], []], [3, [0, 1, 2], []], [4, [0, 2], []], [5, [1], []], [6, [1, 2], []], [7, [2], []]];empty/full opens present;every result in SAME TARGET topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S009
操作前
TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=1,U=[0]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1], [0]], [3, [0, 1, 2], [0]], [4, [0, 2], [0]], [5, [0, 1], []], [6, [0, 1, 2], []], [7, [0, 2], []]];empty/full opens present;every result in SAME TARGET topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S010
操作前
TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=2,U=[0, 1]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [0, 1], []], [1, [0, 1], [0]], [2, [0, 1], [0, 1]], [3, [0, 1, 2], [0, 1]], [4, [0, 1, 2], [0]], [5, [0, 1], [1]], [6, [0, 1, 2], [1]], [7, [0, 1, 2], []]];empty/full opens present;every result in SAME TARGET topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S011
操作前
TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=3,U=[0, 1, 2]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2], []], [1, [0, 1, 2], [0]], [2, [0, 1, 2], [0, 1]], [3, [0, 1, 2], [0, 1, 2]], [4, [0, 1, 2], [0, 2]], [5, [0, 1, 2], [1]], [6, [0, 1, 2], [1, 2]], [7, [0, 1, 2], [2]]];empty/full opens present;every result in SAME TARGET topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S012
操作前
TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=4,U=[0, 2]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [0, 2], []], [1, [0, 2], [0]], [2, [0, 1, 2], [0]], [3, [0, 1, 2], [0, 2]], [4, [0, 2], [0, 2]], [5, [0, 1, 2], []], [6, [0, 1, 2], [2]], [7, [0, 2], [2]]];empty/full opens present;every result in SAME TARGET topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S013
操作前
TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=5,U=[1]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [1], []], [1, [0, 1], []], [2, [0, 1], [1]], [3, [0, 1, 2], [1]], [4, [0, 1, 2], []], [5, [1], [1]], [6, [1, 2], [1]], [7, [1, 2], []]];empty/full opens present;every result in SAME TARGET topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S014
操作前
TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=6,U=[1, 2]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [1, 2], []], [1, [0, 1, 2], []], [2, [0, 1, 2], [1]], [3, [0, 1, 2], [1, 2]], [4, [0, 1, 2], [2]], [5, [1, 2], [1]], [6, [1, 2], [1, 2]], [7, [1, 2], [2]]];empty/full opens present;every result in SAME TARGET topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S015
操作前
TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=7,U=[2]
操作後
ALL [other_open_index,union,intersection] rows=[[0, [2], []], [1, [0, 2], []], [2, [0, 1, 2], []], [3, [0, 1, 2], [2]], [4, [0, 2], [2]], [5, [1, 2], []], [6, [1, 2], [2]], [7, [2], [2]]];empty/full opens present;every result in SAME TARGET topology
條件：無
先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。
規則：compactimg_qualified_complete_source_or_target_topology · 依賴：G001, S016
操作前
ORIGINAL target-open index=0,O=[];total original map=[1, 1, 0];ENTIRE source carrier Fin(3)
操作後
full source preimage=[];source-open=是;NOT restricted to K
條件：無
連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。
規則：compactimg_EVERY_target_open_FULL_source_preimage · 依賴：G001, S017
操作前
ORIGINAL target-open index=1,O=[0];total original map=[1, 1, 0];ENTIRE source carrier Fin(3)
操作後
full source preimage=[2];source-open=是;NOT restricted to K
條件：無
連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。
規則：compactimg_EVERY_target_open_FULL_source_preimage · 依賴：G001, S018
操作前
ORIGINAL target-open index=2,O=[0, 1];total original map=[1, 1, 0];ENTIRE source carrier Fin(3)
操作後
full source preimage=[0, 1, 2];source-open=是;NOT restricted to K
條件：無
連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。
規則：compactimg_EVERY_target_open_FULL_source_preimage · 依賴：G001, S019
操作前
ORIGINAL target-open index=3,O=[0, 1, 2];total original map=[1, 1, 0];ENTIRE source carrier Fin(3)
操作後
full source preimage=[0, 1, 2];source-open=是;NOT restricted to K
條件：無
連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。
規則：compactimg_EVERY_target_open_FULL_source_preimage · 依賴：G001, S020
操作前
ORIGINAL target-open index=4,O=[0, 2];total original map=[1, 1, 0];ENTIRE source carrier Fin(3)
操作後
full source preimage=[2];source-open=是;NOT restricted to K
條件：無
連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。
規則：compactimg_EVERY_target_open_FULL_source_preimage · 依賴：G001, S021
操作前
ORIGINAL target-open index=5,O=[1];total original map=[1, 1, 0];ENTIRE source carrier Fin(3)
操作後
full source preimage=[0, 1];source-open=是;NOT restricted to K
條件：無
連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。
規則：compactimg_EVERY_target_open_FULL_source_preimage · 依賴：G001, S022
操作前
ORIGINAL target-open index=6,O=[1, 2];total original map=[1, 1, 0];ENTIRE source carrier Fin(3)
操作後
full source preimage=[0, 1];source-open=是;NOT restricted to K
條件：無
連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。
規則：compactimg_EVERY_target_open_FULL_source_preimage · 依賴：G001, S023
操作前
ORIGINAL target-open index=7,O=[2];total original map=[1, 1, 0];ENTIRE source carrier Fin(3)
操作後
full source preimage=[];source-open=是;NOT restricted to K
條件：無
連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。
規則：compactimg_EVERY_target_open_FULL_source_preimage · 依賴：G001, S024
操作前
ORIGINAL source subset K=[0, 1, 2];ALL [x,map(x)] pairs=[[0, 1], [1, 1], [2, 0]];target carrier Fin(3)
操作後
exact sorted distinct image=[0, 1];K and image finite hence compact;continuous=是 is the ALL-target-open conjunction
條件：無
每個像由原K中的點見證，反向每個K像均已列，所以集合恰等f(K)。有限集合任意open cover可逐點選一個open給有限子cover；不需先證連續，但一般連續像定理另有拉回證明。
規則：compactimg_finite_image_and_continuity_separate · 依賴：G001, S025
操作前
ORIGINAL indexed TARGET-open cover=[[0], [1]] ACTUALLY covers f(K)=[0, 1];continuous=是
操作後
ALL ORIGINAL-index relative pullbacks K intersect f^-1(O_i)=[[2], [0, 1]];they cover K and are relative-open;compact K gives finite ORIGINAL indices,which cover image again
條件：無
連續性給整個source的open逆像，再交K；cover像才保拉回覆K。任意image點取原K的preimage，有限拉回子cover中的同一索引便覆蓋它；不需injective或surjective到全部target。
規則：compactimg_original_image_cover_relative_pullbacks_general_proof · 依賴：G001, S026
操作前
ORIGINAL cover indices0..1;relative pullbacks=[[2], [0, 1]];test increasing cardinality then NUMERIC increasing bitmask
操作後
ALL tested [mask,indices,covered_K_points,covers_K] rows=[[0, [], [], 否], [1, [0], [2], 否], [2, [1], [0, 1], 否], [3, [0, 1], [0, 1, 2], 是]];FIRST passing selected=[0, 1];selected target opens=[[0], [1]]
條件：無
較少索引的候選全部失敗，同數量中較小numeric mask也都失敗，才取來源首解。這不是索引列表的LEX順序；空K的空選擇合法，不把[]當未找到；最小不代表唯一。
規則：compactimg_minimum_cardinality_then_first_NUMERIC_mask · 依賴：G001, S027
S028 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S024 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S023 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S022 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S017 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S016 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S013 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。([0, 1], [[0], [1]], [[2], [0, 1]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)
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: 5abad99f57ab223ceb8cf5c478b5983f4e5f30721c28586af49c12129766688ecandidate record SHA-256: b67a936a13065ab8dfd959de583f5c048e90eb5175c0e68ae74b270cf6845986semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5eLean: NOT_RUN · human review: NOT_PERFORMED