# Original Open Covers, Indexed Subcovers and Inclusion-Minimality Boundaries

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0001/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 2. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0002/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 3. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0003/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 4. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0004/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 5. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0005/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 6. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0006/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 7. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0007/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 8. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0008/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 9. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0009/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 10. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0010/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 11. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0011/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 12. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0012/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 13. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0013/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 14. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0014/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 15. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0015/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 16. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0016/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 17. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0017/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 18. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0018/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 19. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0019/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 20. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0020/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 21. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0021/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 22. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0022/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 23. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0023/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 24. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0024/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 25. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0025/)

Open cover and subcover · qf-mod32-cover-audit · candidate





## 26. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0026/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 27. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0027/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 28. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0028/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 29. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0029/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 30. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0030/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 31. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0031/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 32. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0032/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 33. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0033/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 34. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0034/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 35. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0035/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 36. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0036/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 37. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0037/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 38. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0038/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 39. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0039/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 40. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0040/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 41. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0041/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 42. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0042/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 43. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0043/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 44. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0044/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 45. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0045/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 46. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0046/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 47. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0047/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 48. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0048/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 49. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0049/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 50. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0050/)

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate





## 51. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0051/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 52. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0052/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 53. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0053/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 54. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0054/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 55. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0055/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 56. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0056/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 57. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0057/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 58. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0058/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 59. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0059/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 60. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0060/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 61. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0061/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 62. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0062/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 63. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0063/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 64. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0064/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 65. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0065/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 66. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0066/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 67. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0067/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 68. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0068/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 69. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0069/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 70. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0070/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 71. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0071/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 72. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0072/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 73. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0073/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 74. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0074/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 75. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0075/)

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate





## 76. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0076/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 77. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0077/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 78. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0078/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 79. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0079/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 80. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0080/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 81. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0081/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 82. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0082/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 83. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0083/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 84. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0084/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 85. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0085/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 86. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0086/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 87. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0087/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 88. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0088/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 89. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0089/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 90. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0090/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 91. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0091/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 92. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0092/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 93. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0093/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 94. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0094/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 95. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0095/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 96. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0096/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 97. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0097/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 98. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0098/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 99. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0099/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate





## 100. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0100/)

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations/validation-report.json)