有限乘積拓撲基礎 · 脈衝 R04
本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。
本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。
batch-20261015-advance-20261117-r04-finite-product-topology-foundations · 2026-10-15 · 100 題
欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。
欄序與來源綁定資料
1. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []], 2, 2, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []], 2, 2, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
2. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
3. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
4. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
5. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
6. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
7. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
8. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
9. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
10. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
11. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
12. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], [[2, 100], [2, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], [[2, 100], [2, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
13. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
14. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
15. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
16. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], [[2, 100], [2, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], [[2, 100], [2, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
17. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], []], 3, 3, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
18. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
19. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
20. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
21. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
22. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
23. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], 6, 6, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], 6, 6, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
24. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
25. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-rectangle-generation；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate] 欄序回答。
查看答案與解釋
([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[2, 100], [2, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)
Wave6第十一層核對=([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[2, 100], [2, 101]], []], 5, 5, 是, rectangles_generate_opens_by_arbitrary_unions)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-rectangle-generation · candidate
26. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
27. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
28. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
29. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
30. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
31. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
32. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
33. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
34. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
35. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
36. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
37. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是], [[0], [[0, 100], [0, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], 是], [[0], [[0, 100], [0, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
38. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
39. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
40. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
41. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是], [[1], [[1, 100], [1, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], 是], [[1], [[1, 100], [1, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
42. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
43. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
44. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
45. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
46. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
47. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
48. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[0], [[0, 100], [0, 101]], 是], [[1], [[1, 100], [1, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
49. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
50. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-projection-continuity；資料={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}。請依 [first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate] 欄序回答。
查看答案與解釋
([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)
Wave6第十一層核對=([[[], [], 是], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], 是], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], 是], [[2], [[2, 100], [2, 101]], 是]], [[[], [], 是], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], 是]], 是, 是, 是, all_projection_preimages_are_open)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-projection-continuity · candidate
51. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
52. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
53. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
54. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
55. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
56. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
57. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
58. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
59. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
60. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
61. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
62. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
63. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
64. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
65. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
66. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
67. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
68. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
69. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
70. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
71. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
72. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
73. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(否, 是, 否, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
74. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0],[1],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
75. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-coordinate-criterion；資料={"z_points":[0,1],"z_topology":[[],[0,1]],"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"map_x":[0,1],"map_y":[100,101]}。請依 [first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate] 欄序回答。
兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。
查看答案與解釋
(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)
Wave6第十一層核對=(是, 是, 是, 是, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-coordinate-criterion · candidate
76. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"nonrectangle_open","model":{"x_points":[0,1],"x_topology":[[],[0],[1],[0,1]],"y_points":[100,101],"y_topology":[[],[100],[101],[100,101]],"candidate_open":[[0,100],[1,101]]}}。請依 [mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(nonrectangle_open, [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [1, 101]], 是, 否, [[[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[0, 100], [1, 100]], [[0, 100]], [[0, 101], [1, 101]], [[0, 101]], [[1, 100], [1, 101]], [[1, 100]], [[1, 101]], []], open_union_of_rectangles_need_not_be_one_rectangle)
Wave6第十一層核對=(nonrectangle_open, [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [1, 101]], 是, 否, [[[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[0, 100], [1, 100]], [[0, 100]], [[0, 101], [1, 101]], [[0, 101]], [[1, 100], [1, 101]], [[1, 100]], [[1, 101]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
77. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"invalid_factor","model":{"x_points":[0,1],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"}}。請依 [mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)
Wave6第十一層核對=(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
78. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"finite_box_no_extrapolation","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}}。請依 [mode, product_point_count, rectangle_count, open_count, finite_model_certificate, finite_box_statement, infinite_extrapolation_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(finite_box_no_extrapolation, 8, 3, 3, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)
Wave6第十一層核對=(finite_box_no_extrapolation, 8, 3, 3, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
79. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"ordered_pair_direction","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]],"pair":[0,101]}}。請依 [mode, pair, reversed_pair, pair_identity_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)
Wave6第十一層核對=(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
80. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"nonrectangle_open","model":{"x_points":[40,41],"x_topology":[[],[40],[41],[40,41]],"y_points":[140,141],"y_topology":[[],[140],[141],[140,141]],"candidate_open":[[40,140],[41,141]]}}。請依 [mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(nonrectangle_open, [[40, 140], [40, 141], [41, 140], [41, 141]], [[40, 140], [41, 141]], 是, 否, [[[40, 140], [40, 141], [41, 140], [41, 141]], [[40, 140], [40, 141]], [[40, 140], [41, 140]], [[40, 140]], [[40, 141], [41, 141]], [[40, 141]], [[41, 140], [41, 141]], [[41, 140]], [[41, 141]], []], open_union_of_rectangles_need_not_be_one_rectangle)
Wave6第十一層核對=(nonrectangle_open, [[40, 140], [40, 141], [41, 140], [41, 141]], [[40, 140], [41, 141]], 是, 否, [[[40, 140], [40, 141], [41, 140], [41, 141]], [[40, 140], [40, 141]], [[40, 140], [41, 140]], [[40, 140]], [[40, 141], [41, 141]], [[40, 141]], [[41, 140], [41, 141]], [[41, 140]], [[41, 141]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
81. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"invalid_factor","model":{"x_points":[0,1,2],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"}}。請依 [mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)
Wave6第十一層核對=(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
82. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"finite_box_no_extrapolation","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}}。請依 [mode, product_point_count, rectangle_count, open_count, finite_model_certificate, finite_box_statement, infinite_extrapolation_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(finite_box_no_extrapolation, 8, 5, 5, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)
Wave6第十一層核對=(finite_box_no_extrapolation, 8, 5, 5, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
83. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"ordered_pair_direction","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"pair":[0,101]}}。請依 [mode, pair, reversed_pair, pair_identity_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)
Wave6第十一層核對=(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
84. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"nonrectangle_open","model":{"x_points":[80,81],"x_topology":[[],[80],[81],[80,81]],"y_points":[180,181],"y_topology":[[],[180],[181],[180,181]],"candidate_open":[[80,180],[81,181]]}}。請依 [mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(nonrectangle_open, [[80, 180], [80, 181], [81, 180], [81, 181]], [[80, 180], [81, 181]], 是, 否, [[[80, 180], [80, 181], [81, 180], [81, 181]], [[80, 180], [80, 181]], [[80, 180], [81, 180]], [[80, 180]], [[80, 181], [81, 181]], [[80, 181]], [[81, 180], [81, 181]], [[81, 180]], [[81, 181]], []], open_union_of_rectangles_need_not_be_one_rectangle)
Wave6第十一層核對=(nonrectangle_open, [[80, 180], [80, 181], [81, 180], [81, 181]], [[80, 180], [81, 181]], 是, 否, [[[80, 180], [80, 181], [81, 180], [81, 181]], [[80, 180], [80, 181]], [[80, 180], [81, 180]], [[80, 180]], [[80, 181], [81, 181]], [[80, 181]], [[81, 180], [81, 181]], [[81, 180]], [[81, 181]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
85. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"invalid_factor","model":{"x_points":[0,1,2,3],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"}}。請依 [mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)
Wave6第十一層核對=(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
86. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"finite_box_no_extrapolation","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}}。請依 [mode, product_point_count, rectangle_count, open_count, finite_model_certificate, finite_box_statement, infinite_extrapolation_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(finite_box_no_extrapolation, 8, 4, 4, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)
Wave6第十一層核對=(finite_box_no_extrapolation, 8, 4, 4, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
87. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"ordered_pair_direction","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,2],[0],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"pair":[0,101]}}。請依 [mode, pair, reversed_pair, pair_identity_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)
Wave6第十一層核對=(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
88. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"nonrectangle_open","model":{"x_points":[120,121],"x_topology":[[],[120],[121],[120,121]],"y_points":[220,221],"y_topology":[[],[220],[221],[220,221]],"candidate_open":[[120,220],[121,221]]}}。請依 [mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(nonrectangle_open, [[120, 220], [120, 221], [121, 220], [121, 221]], [[120, 220], [121, 221]], 是, 否, [[[120, 220], [120, 221], [121, 220], [121, 221]], [[120, 220], [120, 221]], [[120, 220], [121, 220]], [[120, 220]], [[120, 221], [121, 221]], [[120, 221]], [[121, 220], [121, 221]], [[121, 220]], [[121, 221]], []], open_union_of_rectangles_need_not_be_one_rectangle)
Wave6第十一層核對=(nonrectangle_open, [[120, 220], [120, 221], [121, 220], [121, 221]], [[120, 220], [121, 221]], 是, 否, [[[120, 220], [120, 221], [121, 220], [121, 221]], [[120, 220], [120, 221]], [[120, 220], [121, 220]], [[120, 220]], [[120, 221], [121, 221]], [[120, 221]], [[121, 220], [121, 221]], [[121, 220]], [[121, 221]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
89. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"invalid_factor","model":{"x_points":[0,1,2,3,4],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"}}。請依 [mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)
Wave6第十一層核對=(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
90. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"finite_box_no_extrapolation","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]]}}。請依 [mode, product_point_count, rectangle_count, open_count, finite_model_certificate, finite_box_statement, infinite_extrapolation_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(finite_box_no_extrapolation, 8, 4, 4, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)
Wave6第十一層核對=(finite_box_no_extrapolation, 8, 4, 4, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
91. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"ordered_pair_direction","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[1,2],[1],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"pair":[0,101]}}。請依 [mode, pair, reversed_pair, pair_identity_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)
Wave6第十一層核對=(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
92. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"nonrectangle_open","model":{"x_points":[160,161],"x_topology":[[],[160],[161],[160,161]],"y_points":[260,261],"y_topology":[[],[260],[261],[260,261]],"candidate_open":[[160,260],[161,261]]}}。請依 [mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(nonrectangle_open, [[160, 260], [160, 261], [161, 260], [161, 261]], [[160, 260], [161, 261]], 是, 否, [[[160, 260], [160, 261], [161, 260], [161, 261]], [[160, 260], [160, 261]], [[160, 260], [161, 260]], [[160, 260]], [[160, 261], [161, 261]], [[160, 261]], [[161, 260], [161, 261]], [[161, 260]], [[161, 261]], []], open_union_of_rectangles_need_not_be_one_rectangle)
Wave6第十一層核對=(nonrectangle_open, [[160, 260], [160, 261], [161, 260], [161, 261]], [[160, 260], [161, 261]], 是, 否, [[[160, 260], [160, 261], [161, 260], [161, 261]], [[160, 260], [160, 261]], [[160, 260], [161, 260]], [[160, 260]], [[160, 261], [161, 261]], [[160, 261]], [[161, 260], [161, 261]], [[161, 260]], [[161, 261]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
93. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"invalid_factor","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"}}。請依 [mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)
Wave6第十一層核對=(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
94. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"finite_box_no_extrapolation","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}}。請依 [mode, product_point_count, rectangle_count, open_count, finite_model_certificate, finite_box_statement, infinite_extrapolation_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(finite_box_no_extrapolation, 8, 4, 4, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)
Wave6第十一層核對=(finite_box_no_extrapolation, 8, 4, 4, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
95. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"ordered_pair_direction","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1]],"y_points":[100,101],"y_topology":[[],[100,101]],"pair":[0,101]}}。請依 [mode, pair, reversed_pair, pair_identity_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)
Wave6第十一層核對=(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
96. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"nonrectangle_open","model":{"x_points":[200,201],"x_topology":[[],[200],[201],[200,201]],"y_points":[300,301],"y_topology":[[],[300],[301],[300,301]],"candidate_open":[[200,300],[201,301]]}}。請依 [mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(nonrectangle_open, [[200, 300], [200, 301], [201, 300], [201, 301]], [[200, 300], [201, 301]], 是, 否, [[[200, 300], [200, 301], [201, 300], [201, 301]], [[200, 300], [200, 301]], [[200, 300], [201, 300]], [[200, 300]], [[200, 301], [201, 301]], [[200, 301]], [[201, 300], [201, 301]], [[201, 300]], [[201, 301]], []], open_union_of_rectangles_need_not_be_one_rectangle)
Wave6第十一層核對=(nonrectangle_open, [[200, 300], [200, 301], [201, 300], [201, 301]], [[200, 300], [201, 301]], 是, 否, [[[200, 300], [200, 301], [201, 300], [201, 301]], [[200, 300], [200, 301]], [[200, 300], [201, 300]], [[200, 300]], [[200, 301], [201, 301]], [[200, 301]], [[201, 300], [201, 301]], [[201, 300]], [[201, 301]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
97. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"invalid_factor","model":{"x_points":[0,1,2,3,4,5,6],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"}}。請依 [mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)
Wave6第十一層核對=(invalid_factor, x, 否, empty and full opens required, 是, null, 否, product_topology_requires_valid_factor_topologies)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
98. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"finite_box_no_extrapolation","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],"y_points":[100,101],"y_topology":[[],[100,101]]}}。請依 [mode, product_point_count, rectangle_count, open_count, finite_model_certificate, finite_box_statement, infinite_extrapolation_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(finite_box_no_extrapolation, 8, 6, 6, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)
Wave6第十一層核對=(finite_box_no_extrapolation, 8, 6, 6, 是, finite_product_and_box_topologies_coincide, 否, no_inference_to_infinite_products)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
99. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"ordered_pair_direction","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2,3],[0,1,2],[2]],"y_points":[100,101],"y_topology":[[],[100,101]],"pair":[0,101]}}。請依 [mode, pair, reversed_pair, pair_identity_claim, boundary] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)
Wave6第十一層核對=(ordered_pair_direction, [0, 101], [101, 0], 否, ordered_pair_coordinates_are_typed_and_directional)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
100. 有限乘積拓撲
完整finite factor topologies、Cartesian carrier與rectangle-union生成責任。操作=product-topology-boundary；資料={"mode":"nonrectangle_open","model":{"x_points":[240,241],"x_topology":[[],[240],[241],[240,241]],"y_points":[340,341],"y_topology":[[],[340],[341],[340,341]],"candidate_open":[[240,340],[241,341]]}}。請依 [mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate] 欄序回答。
四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。
查看答案與解釋
(nonrectangle_open, [[240, 340], [240, 341], [241, 340], [241, 341]], [[240, 340], [241, 341]], 是, 否, [[[240, 340], [240, 341], [241, 340], [241, 341]], [[240, 340], [240, 341]], [[240, 340], [241, 340]], [[240, 340]], [[240, 341], [241, 341]], [[240, 341]], [[241, 340], [241, 341]], [[241, 340]], [[241, 341]], []], open_union_of_rectangles_need_not_be_one_rectangle)
Wave6第十一層核對=(nonrectangle_open, [[240, 340], [240, 341], [241, 340], [241, 341]], [[240, 340], [241, 341]], 是, 否, [[[240, 340], [240, 341], [241, 340], [241, 341]], [[240, 340], [240, 341]], [[240, 340], [241, 340]], [[240, 340]], [[240, 341], [241, 341]], [[240, 341]], [[241, 340], [241, 341]], [[241, 340]], [[241, 341]], []], open_union_of_rectangles_need_not_be_one_rectangle)；依完整明示資料獨立重算；projections與coordinate criterion在完整product topology上核對；basic rectangles不等於全部opens；finite product等於box不得外推到infinite product。
本題逐步詳解（AI 二次覆核）
有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate
原始題目 JSONL · 驗收報告