第 19010 題：同胚

原題與已知條件

完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,2,3,4,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,2,3,4,1]}。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-homeomorphism

逐步推導

source operation=homeo-continuous-bijection-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary]

original complete carrier Fin(5);original TOTAL source-indexed candidate map=[0, 2, 3, 4, 1]

all target values distinct and complete;explicit inverse=[0, 4, 1, 2, 3];inverse(map(x))=x and map(inverse(y))=y for EVERY carrier point

source topology:complete original supplied topology family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]];carrier Fin(5)

internally sorted subsets without deduplication=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]];canonical whole-family key order retained;empty and FULL carrier present

source topology:left open index=0;left set=[];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[0, 0], [1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0]]

source topology:left open index=1;left set=[0];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[1, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [2, 0], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0]]

source topology:left open index=2;left set=[0, 1];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[2, 0], [2, 1], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [2, 17], [3, 17], [4, 17], [5, 17], [6, 17], [7, 17], [8, 17], [9, 17], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0]]

source topology:left open index=3;left set=[0, 1, 2];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[3, 0], [3, 1], [3, 2], [3, 3], [4, 3], [5, 3], [6, 3], [4, 2], [5, 2], [6, 2], [3, 10], [4, 10], [5, 10], [6, 10], [4, 1], [5, 1], [6, 1], [3, 17], [3, 18], [4, 18], [5, 18], [6, 18], [4, 17], [5, 17], [6, 17], [3, 25], [4, 25], [5, 25], [6, 25], [4, 0], [5, 0], [6, 0]]

source topology:left open index=4;left set=[0, 1, 2, 3];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[4, 0], [4, 1], [4, 2], [4, 3], [4, 4], [5, 4], [5, 3], [4, 7], [5, 7], [5, 2], [4, 10], [4, 11], [5, 11], [5, 10], [4, 14], [5, 14], [5, 1], [4, 17], [4, 18], [4, 19], [5, 19], [5, 18], [4, 22], [5, 22], [5, 17], [4, 25], [4, 26], [5, 26], [5, 25], [4, 29], [5, 29], [5, 0]]

source topology:left open index=5;left set=[0, 1, 2, 3, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[5, 0], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5], [5, 6], [5, 7], [5, 8], [5, 9], [5, 10], [5, 11], [5, 12], [5, 13], [5, 14], [5, 15], [5, 16], [5, 17], [5, 18], [5, 19], [5, 20], [5, 21], [5, 22], [5, 23], [5, 24], [5, 25], [5, 26], [5, 27], [5, 28], [5, 29], [5, 30], [5, 31]]

source topology:left open index=6;left set=[0, 1, 2, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[6, 0], [6, 1], [6, 2], [6, 3], [5, 3], [5, 6], [6, 6], [5, 2], [5, 9], [6, 9], [6, 10], [5, 10], [5, 13], [6, 13], [5, 1], [5, 16], [6, 16], [6, 17], [6, 18], [5, 18], [5, 21], [6, 21], [5, 17], [5, 24], [6, 24], [6, 25], [5, 25], [5, 28], [6, 28], [5, 0], [5, 31], [6, 31]]

source topology:left open index=7;left set=[0, 1, 3];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[7, 0], [7, 1], [7, 2], [4, 2], [4, 7], [5, 7], [5, 2], [7, 7], [8, 7], [8, 2], [4, 1], [4, 14], [5, 14], [5, 1], [7, 14], [8, 14], [8, 1], [7, 17], [4, 17], [4, 22], [5, 22], [5, 17], [7, 22], [8, 22], [8, 17], [4, 0], [4, 29], [5, 29], [5, 0], [7, 29], [8, 29], [8, 0]]

source topology:left open index=8;left set=[0, 1, 3, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[8, 0], [8, 1], [8, 2], [5, 2], [5, 7], [5, 8], [5, 9], [8, 7], [8, 8], [8, 9], [5, 1], [5, 14], [5, 15], [5, 16], [8, 14], [8, 15], [8, 16], [8, 17], [5, 17], [5, 22], [5, 23], [5, 24], [8, 22], [8, 23], [8, 24], [5, 0], [5, 29], [5, 30], [5, 31], [8, 29], [8, 30], [8, 31]]

source topology:left open index=9;left set=[0, 1, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[9, 0], [9, 1], [9, 2], [6, 2], [5, 2], [5, 9], [6, 9], [8, 2], [8, 9], [9, 9], [6, 1], [5, 1], [5, 16], [6, 16], [8, 1], [8, 16], [9, 16], [9, 17], [6, 17], [5, 17], [5, 24], [6, 24], [8, 17], [8, 24], [9, 24], [6, 0], [5, 0], [5, 31], [6, 31], [8, 0], [8, 31], [9, 31]]

source topology:left open index=10;left set=[0, 2];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[10, 0], [10, 1], [3, 1], [3, 10], [4, 10], [5, 10], [6, 10], [4, 1], [5, 1], [6, 1], [10, 10], [11, 10], [12, 10], [13, 10], [11, 1], [12, 1], [13, 1], [3, 0], [3, 25], [4, 25], [5, 25], [6, 25], [4, 0], [5, 0], [6, 0], [10, 25], [11, 25], [12, 25], [13, 25], [11, 0], [12, 0], [13, 0]]

source topology:left open index=11;left set=[0, 2, 3];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[11, 0], [11, 1], [4, 1], [4, 10], [4, 11], [5, 11], [5, 10], [4, 14], [5, 14], [5, 1], [11, 10], [11, 11], [12, 11], [12, 10], [11, 14], [12, 14], [12, 1], [4, 0], [4, 25], [4, 26], [5, 26], [5, 25], [4, 29], [5, 29], [5, 0], [11, 25], [11, 26], [12, 26], [12, 25], [11, 29], [12, 29], [12, 0]]

source topology:left open index=12;left set=[0, 2, 3, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[12, 0], [12, 1], [5, 1], [5, 10], [5, 11], [5, 12], [5, 13], [5, 14], [5, 15], [5, 16], [12, 10], [12, 11], [12, 12], [12, 13], [12, 14], [12, 15], [12, 16], [5, 0], [5, 25], [5, 26], [5, 27], [5, 28], [5, 29], [5, 30], [5, 31], [12, 25], [12, 26], [12, 27], [12, 28], [12, 29], [12, 30], [12, 31]]

source topology:left open index=13;left set=[0, 2, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[13, 0], [13, 1], [6, 1], [6, 10], [5, 10], [5, 13], [6, 13], [5, 1], [5, 16], [6, 16], [13, 10], [12, 10], [12, 13], [13, 13], [12, 1], [12, 16], [13, 16], [6, 0], [6, 25], [5, 25], [5, 28], [6, 28], [5, 0], [5, 31], [6, 31], [13, 25], [12, 25], [12, 28], [13, 28], [12, 0], [12, 31], [13, 31]]

source topology:left open index=14;left set=[0, 3];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[14, 0], [14, 1], [7, 1], [4, 1], [4, 14], [5, 14], [5, 1], [7, 14], [8, 14], [8, 1], [11, 1], [11, 14], [12, 14], [12, 1], [14, 14], [15, 14], [15, 1], [7, 0], [4, 0], [4, 29], [5, 29], [5, 0], [7, 29], [8, 29], [8, 0], [11, 0], [11, 29], [12, 29], [12, 0], [14, 29], [15, 29], [15, 0]]

source topology:left open index=15;left set=[0, 3, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[15, 0], [15, 1], [8, 1], [5, 1], [5, 14], [5, 15], [5, 16], [8, 14], [8, 15], [8, 16], [12, 1], [12, 14], [12, 15], [12, 16], [15, 14], [15, 15], [15, 16], [8, 0], [5, 0], [5, 29], [5, 30], [5, 31], [8, 29], [8, 30], [8, 31], [12, 0], [12, 29], [12, 30], [12, 31], [15, 29], [15, 30], [15, 31]]

source topology:left open index=16;left set=[0, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[16, 0], [16, 1], [9, 1], [6, 1], [5, 1], [5, 16], [6, 16], [8, 1], [8, 16], [9, 16], [13, 1], [12, 1], [12, 16], [13, 16], [15, 1], [15, 16], [16, 16], [9, 0], [6, 0], [5, 0], [5, 31], [6, 31], [8, 0], [8, 31], [9, 31], [13, 0], [12, 0], [12, 31], [13, 31], [15, 0], [15, 31], [16, 31]]

source topology:left open index=17;left set=[1];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[17, 0], [2, 0], [2, 17], [3, 17], [4, 17], [5, 17], [6, 17], [7, 17], [8, 17], [9, 17], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [17, 17], [18, 17], [19, 17], [20, 17], [21, 17], [22, 17], [23, 17], [24, 17], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0]]

source topology:left open index=18;left set=[1, 2];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[18, 0], [3, 0], [3, 17], [3, 18], [4, 18], [5, 18], [6, 18], [4, 17], [5, 17], [6, 17], [3, 25], [4, 25], [5, 25], [6, 25], [4, 0], [5, 0], [6, 0], [18, 17], [18, 18], [19, 18], [20, 18], [21, 18], [19, 17], [20, 17], [21, 17], [18, 25], [19, 25], [20, 25], [21, 25], [19, 0], [20, 0], [21, 0]]

source topology:left open index=19;left set=[1, 2, 3];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[19, 0], [4, 0], [4, 17], [4, 18], [4, 19], [5, 19], [5, 18], [4, 22], [5, 22], [5, 17], [4, 25], [4, 26], [5, 26], [5, 25], [4, 29], [5, 29], [5, 0], [19, 17], [19, 18], [19, 19], [20, 19], [20, 18], [19, 22], [20, 22], [20, 17], [19, 25], [19, 26], [20, 26], [20, 25], [19, 29], [20, 29], [20, 0]]

source topology:left open index=20;left set=[1, 2, 3, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[20, 0], [5, 0], [5, 17], [5, 18], [5, 19], [5, 20], [5, 21], [5, 22], [5, 23], [5, 24], [5, 25], [5, 26], [5, 27], [5, 28], [5, 29], [5, 30], [5, 31], [20, 17], [20, 18], [20, 19], [20, 20], [20, 21], [20, 22], [20, 23], [20, 24], [20, 25], [20, 26], [20, 27], [20, 28], [20, 29], [20, 30], [20, 31]]

source topology:left open index=21;left set=[1, 2, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[21, 0], [6, 0], [6, 17], [6, 18], [5, 18], [5, 21], [6, 21], [5, 17], [5, 24], [6, 24], [6, 25], [5, 25], [5, 28], [6, 28], [5, 0], [5, 31], [6, 31], [21, 17], [21, 18], [20, 18], [20, 21], [21, 21], [20, 17], [20, 24], [21, 24], [21, 25], [20, 25], [20, 28], [21, 28], [20, 0], [20, 31], [21, 31]]

source topology:left open index=22;left set=[1, 3];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[22, 0], [7, 0], [7, 17], [4, 17], [4, 22], [5, 22], [5, 17], [7, 22], [8, 22], [8, 17], [4, 0], [4, 29], [5, 29], [5, 0], [7, 29], [8, 29], [8, 0], [22, 17], [19, 17], [19, 22], [20, 22], [20, 17], [22, 22], [23, 22], [23, 17], [19, 0], [19, 29], [20, 29], [20, 0], [22, 29], [23, 29], [23, 0]]

source topology:left open index=23;left set=[1, 3, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[23, 0], [8, 0], [8, 17], [5, 17], [5, 22], [5, 23], [5, 24], [8, 22], [8, 23], [8, 24], [5, 0], [5, 29], [5, 30], [5, 31], [8, 29], [8, 30], [8, 31], [23, 17], [20, 17], [20, 22], [20, 23], [20, 24], [23, 22], [23, 23], [23, 24], [20, 0], [20, 29], [20, 30], [20, 31], [23, 29], [23, 30], [23, 31]]

source topology:left open index=24;left set=[1, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[24, 0], [9, 0], [9, 17], [6, 17], [5, 17], [5, 24], [6, 24], [8, 17], [8, 24], [9, 24], [6, 0], [5, 0], [5, 31], [6, 31], [8, 0], [8, 31], [9, 31], [24, 17], [21, 17], [20, 17], [20, 24], [21, 24], [23, 17], [23, 24], [24, 24], [21, 0], [20, 0], [20, 31], [21, 31], [23, 0], [23, 31], [24, 31]]

source topology:left open index=25;left set=[2];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[25, 0], [10, 0], [3, 0], [3, 25], [4, 25], [5, 25], [6, 25], [4, 0], [5, 0], [6, 0], [10, 25], [11, 25], [12, 25], [13, 25], [11, 0], [12, 0], [13, 0], [18, 0], [18, 25], [19, 25], [20, 25], [21, 25], [19, 0], [20, 0], [21, 0], [25, 25], [26, 25], [27, 25], [28, 25], [26, 0], [27, 0], [28, 0]]

source topology:left open index=26;left set=[2, 3];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[26, 0], [11, 0], [4, 0], [4, 25], [4, 26], [5, 26], [5, 25], [4, 29], [5, 29], [5, 0], [11, 25], [11, 26], [12, 26], [12, 25], [11, 29], [12, 29], [12, 0], [19, 0], [19, 25], [19, 26], [20, 26], [20, 25], [19, 29], [20, 29], [20, 0], [26, 25], [26, 26], [27, 26], [27, 25], [26, 29], [27, 29], [27, 0]]

source topology:left open index=27;left set=[2, 3, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[27, 0], [12, 0], [5, 0], [5, 25], [5, 26], [5, 27], [5, 28], [5, 29], [5, 30], [5, 31], [12, 25], [12, 26], [12, 27], [12, 28], [12, 29], [12, 30], [12, 31], [20, 0], [20, 25], [20, 26], [20, 27], [20, 28], [20, 29], [20, 30], [20, 31], [27, 25], [27, 26], [27, 27], [27, 28], [27, 29], [27, 30], [27, 31]]

source topology:left open index=28;left set=[2, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[28, 0], [13, 0], [6, 0], [6, 25], [5, 25], [5, 28], [6, 28], [5, 0], [5, 31], [6, 31], [13, 25], [12, 25], [12, 28], [13, 28], [12, 0], [12, 31], [13, 31], [21, 0], [21, 25], [20, 25], [20, 28], [21, 28], [20, 0], [20, 31], [21, 31], [28, 25], [27, 25], [27, 28], [28, 28], [27, 0], [27, 31], [28, 31]]

source topology:left open index=29;left set=[3];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[29, 0], [14, 0], [7, 0], [4, 0], [4, 29], [5, 29], [5, 0], [7, 29], [8, 29], [8, 0], [11, 0], [11, 29], [12, 29], [12, 0], [14, 29], [15, 29], [15, 0], [22, 0], [19, 0], [19, 29], [20, 29], [20, 0], [22, 29], [23, 29], [23, 0], [26, 0], [26, 29], [27, 29], [27, 0], [29, 29], [30, 29], [30, 0]]

source topology:left open index=30;left set=[3, 4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[30, 0], [15, 0], [8, 0], [5, 0], [5, 29], [5, 30], [5, 31], [8, 29], [8, 30], [8, 31], [12, 0], [12, 29], [12, 30], [12, 31], [15, 29], [15, 30], [15, 31], [23, 0], [20, 0], [20, 29], [20, 30], [20, 31], [23, 29], [23, 30], [23, 31], [27, 0], [27, 29], [27, 30], [27, 31], [30, 29], [30, 30], [30, 31]]

source topology:left open index=31;left set=[4];ALL right open indices in source family order0..31

ordered [union_open_index,intersection_open_index] for EVERY right open=[[31, 0], [16, 0], [9, 0], [6, 0], [5, 0], [5, 31], [6, 31], [8, 0], [8, 31], [9, 31], [13, 0], [12, 0], [12, 31], [13, 31], [15, 0], [15, 31], [16, 31], [24, 0], [21, 0], [20, 0], [20, 31], [21, 31], [23, 0], [23, 31], [24, 31], [28, 0], [27, 0], [27, 31], [28, 31], [30, 0], [30, 31], [31, 31]]

source topology:empty/full carrier and EVERY binary union/intersection checked in COMPLETE finite family of32 opens

ANY arbitrary indexed union has only finitely many DISTINCT opens here,so equals a finite repeated binary union;empty union is empty;finite intersections include empty intersection=full carrier;all topology axioms hold

target topology:complete original supplied topology family=[[], [0, 1, 2, 3, 4]];carrier Fin(5)

internally sorted subsets without deduplication=[[], [0, 1, 2, 3, 4]];canonical whole-family key order retained;empty and FULL carrier present

target topology:left open index=0;left set=[];ALL right open indices in source family order0..1

ordered [union_open_index,intersection_open_index] for EVERY right open=[[0, 0], [1, 0]]

target topology:left open index=1;left set=[0, 1, 2, 3, 4];ALL right open indices in source family order0..1

ordered [union_open_index,intersection_open_index] for EVERY right open=[[1, 0], [1, 1]]

target topology:empty/full carrier and EVERY binary union/intersection checked in COMPLETE finite family of2 opens

ANY arbitrary indexed union has only finitely many DISTINCT opens here,so equals a finite repeated binary union;empty union is empty;finite intersections include empty intersection=full carrier;all topology axioms hold

forward map:ACTUAL total map=[0, 2, 3, 4, 1];target open index=0;complete target open=[]

exact full-carrier preimage=[];matching source-open index=0;preimage open=是

forward map:ACTUAL total map=[0, 2, 3, 4, 1];target open index=1;complete target open=[0, 1, 2, 3, 4]

exact full-carrier preimage=[0, 1, 2, 3, 4];matching source-open index=5;preimage open=是

forward map:ALL target-open preimage flags=[是, 是]

continuous=是 by the COMPLETE open-preimage definition

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=0;complete target open=[]

exact full-carrier preimage=[];matching source-open index=0;preimage open=是

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=1;complete target open=[0]

exact full-carrier preimage=[0];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=2;complete target open=[0, 1]

exact full-carrier preimage=[0, 2];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=3;complete target open=[0, 1, 2]

exact full-carrier preimage=[0, 2, 3];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=4;complete target open=[0, 1, 2, 3]

exact full-carrier preimage=[0, 2, 3, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=5;complete target open=[0, 1, 2, 3, 4]

exact full-carrier preimage=[0, 1, 2, 3, 4];matching source-open index=1;preimage open=是

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=6;complete target open=[0, 1, 2, 4]

exact full-carrier preimage=[0, 1, 2, 3];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=7;complete target open=[0, 1, 3]

exact full-carrier preimage=[0, 2, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=8;complete target open=[0, 1, 3, 4]

exact full-carrier preimage=[0, 1, 2, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=9;complete target open=[0, 1, 4]

exact full-carrier preimage=[0, 1, 2];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=10;complete target open=[0, 2]

exact full-carrier preimage=[0, 3];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=11;complete target open=[0, 2, 3]

exact full-carrier preimage=[0, 3, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=12;complete target open=[0, 2, 3, 4]

exact full-carrier preimage=[0, 1, 3, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=13;complete target open=[0, 2, 4]

exact full-carrier preimage=[0, 1, 3];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=14;complete target open=[0, 3]

exact full-carrier preimage=[0, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=15;complete target open=[0, 3, 4]

exact full-carrier preimage=[0, 1, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=16;complete target open=[0, 4]

exact full-carrier preimage=[0, 1];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=17;complete target open=[1]

exact full-carrier preimage=[2];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=18;complete target open=[1, 2]

exact full-carrier preimage=[2, 3];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=19;complete target open=[1, 2, 3]

exact full-carrier preimage=[2, 3, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=20;complete target open=[1, 2, 3, 4]

exact full-carrier preimage=[1, 2, 3, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=21;complete target open=[1, 2, 4]

exact full-carrier preimage=[1, 2, 3];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=22;complete target open=[1, 3]

exact full-carrier preimage=[2, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=23;complete target open=[1, 3, 4]

exact full-carrier preimage=[1, 2, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=24;complete target open=[1, 4]

exact full-carrier preimage=[1, 2];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=25;complete target open=[2]

exact full-carrier preimage=[3];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=26;complete target open=[2, 3]

exact full-carrier preimage=[3, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=27;complete target open=[2, 3, 4]

exact full-carrier preimage=[1, 3, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=28;complete target open=[2, 4]

exact full-carrier preimage=[1, 3];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=29;complete target open=[3]

exact full-carrier preimage=[4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=30;complete target open=[3, 4]

exact full-carrier preimage=[1, 4];matching source-open index=null;preimage open=否

inverse map:ACTUAL total map=[0, 4, 1, 2, 3];target open index=31;complete target open=[4]

exact full-carrier preimage=[1];matching source-open index=null;preimage open=否

inverse map:ALL target-open preimage flags=[是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]

continuous=否 by the COMPLETE open-preimage definition

complete valid source/target topologies;actual bijection=[0, 2, 3, 4, 1];forward continuity=是;inverse continuity=否

homeomorphism=否 is the CONJUNCTION;family name does not assume either continuity direction

逐步反向核對

• S077 · 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。
• S076 · 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。
• S075 · 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。
• S074 · 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。
• S073 · 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。
• S072 · 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。
• S071 · 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。
• S070 · 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。
• S069 · 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。
• S068 · 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。
• S067 · 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。
• S066 · 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。
• S065 · 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。
• S064 · 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。
• S063 · 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。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• S058 · 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。
• S057 · 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。
• S056 · 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。
• S055 · 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。
• S054 · 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。
• S053 · 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。
• S052 · 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。
• S051 · 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。
• S050 · 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。
• S049 · 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。
• S048 · 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。
• S047 · 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。
• S046 · 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。
• S045 · 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。
• S044 · 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。
• S043 · 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。
• S042 · 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。
• S041 · 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。
• S040 · 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。
• S039 · 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。
• S038 · 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。
• S037 · 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。
• S036 · 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。
• S035 · 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。
• S034 · 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。
• S033 · 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。
• S032 · 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。
• S031 · 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。
• S030 · 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。
• S029 · 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。
• 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。

結論

(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)

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。