# 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](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0001/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0002/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0003/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0004/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0005/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0006/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0007/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0008/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0009/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0010/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0011/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0012/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0013/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0014/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0015/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0016/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0017/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0018/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0019/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0020/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0021/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0022/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0023/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0024/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0025/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0026/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0027/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0028/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0029/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0030/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0031/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0032/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0033/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0034/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0035/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0036/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0037/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0038/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0039/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0040/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0041/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0042/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0043/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0044/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0045/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0046/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0047/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0048/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0049/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0050/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0051/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0052/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0053/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0054/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0055/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0056/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0057/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0058/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0059/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0060/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0061/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0062/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0063/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0064/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0065/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0066/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0067/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0068/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0069/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0070/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0071/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0072/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0073/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0074/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0075/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0076/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0077/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0078/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0079/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0080/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0081/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0082/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0083/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0084/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0085/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0086/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0087/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0088/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0089/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0090/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0091/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0092/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0093/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0094/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0095/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0096/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0097/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0098/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0099/)

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）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261117-r04-finite-product-topology-0100/)

有限乘積拓撲 · qf-mod48-product-topology-boundary · candidate



[原始題目 JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261117-r04-finite-product-topology-foundations/questions.jsonl) · [驗收報告](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261117-r04-finite-product-topology-foundations/validation-report.json)