# Inclusion-Maximal Connected Subspaces and Finite Clopen Partitions

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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-20261015-advance-20261117-r07-topological-connected-component-foundations · 2026-10-15 · 100 questions

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

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1], [2, 3, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1]], [0, 1], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0001/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 2. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2], [1, 3, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 2]], [0, 2], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0002/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 3. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2], [3, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 2], [0, 1, 2]], [0, 1, 2], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0003/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 4. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3], [1, 2, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 3]], [0, 3], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0004/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 5. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3], [2, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 3], [0, 1, 3]], [0, 1, 3], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0005/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 6. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3], [1, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 2], [0, 3], [0, 2, 3]], [0, 2, 3], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0006/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 7. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3], [4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]], [0, 1, 2, 3], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0007/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 8. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4], [1, 2, 3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 4]], [0, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0008/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 9. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4], [2, 3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 4], [0, 1, 4]], [0, 1, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0009/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 10. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 4], [1, 3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 2], [0, 4], [0, 2, 4]], [0, 2, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0010/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 11. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4], [3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4]], [0, 1, 2, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0011/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 12. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 4], [1, 2, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 3], [0, 4], [0, 3, 4]], [0, 3, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0012/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 13. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4], [2, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 4], [0, 1, 4], [0, 3, 4], [0, 1, 3, 4]], [0, 1, 3, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0013/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 14. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4], [1, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 2], [0, 3], [0, 2, 3], [0, 4], [0, 2, 4], [0, 3, 4], [0, 2, 3, 4]], [0, 2, 3, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0014/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 15. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4], [5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

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

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4], [0, 3, 4], [0, 1, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], [0, 1, 2, 3, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0015/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 16. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 5], [1, 2, 3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

View answer and explanation(0, [[0], [0, 5]], [0, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 5]], [0, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0016/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 17. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 5], [2, 3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

View answer and explanation(0, [[0], [0, 1], [0, 5], [0, 1, 5]], [0, 1, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 5], [0, 1, 5]], [0, 1, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0017/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 18. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 5], [1, 3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

View answer and explanation(0, [[0], [0, 2], [0, 5], [0, 2, 5]], [0, 2, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 2], [0, 5], [0, 2, 5]], [0, 2, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0018/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 19. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5], [3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

View answer and explanation(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5]], [0, 1, 2, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5]], [0, 1, 2, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0019/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 20. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 5], [1, 2, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

View answer and explanation(0, [[0], [0, 3], [0, 5], [0, 3, 5]], [0, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 3], [0, 5], [0, 3, 5]], [0, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0020/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 21. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5], [2, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

View answer and explanation(0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], [0, 1, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], [0, 1, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0021/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 22. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5], [1, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

View answer and explanation(0, [[0], [0, 2], [0, 3], [0, 2, 3], [0, 5], [0, 2, 5], [0, 3, 5], [0, 2, 3, 5]], [0, 2, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 2], [0, 3], [0, 2, 3], [0, 5], [0, 2, 5], [0, 3, 5], [0, 2, 3, 5]], [0, 2, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0022/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 23. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5], [4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

View answer and explanation(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5], [0, 3, 5], [0, 1, 3, 5], [0, 2, 3, 5], [0, 1, 2, 3, 5]], [0, 1, 2, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5], [0, 3, 5], [0, 1, 3, 5], [0, 2, 3, 5], [0, 1, 2, 3, 5]], [0, 1, 2, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0023/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 24. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4, 5], [1, 2, 3]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

View answer and explanation(0, [[0], [0, 4], [0, 5], [0, 4, 5]], [0, 4, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 4], [0, 5], [0, 4, 5]], [0, 4, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0024/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 25. The Union of All Connected Subsets through a Basepoint Is Its Component

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5], [2, 3]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).

View answer and explanation(0, [[0], [0, 1], [0, 4], [0, 1, 4], [0, 5], [0, 1, 5], [0, 4, 5], [0, 1, 4, 5]], [0, 1, 4, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)

Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 4], [0, 1, 4], [0, 5], [0, 1, 5], [0, 4, 5], [0, 1, 4, 5]], [0, 1, 4, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0025/)

Connected component of a topological space · qf-mod48-component-connected-subsets · candidate





## 26. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1], [2, 3, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1], [[0], [0, 1]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0026/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 27. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2], [1, 3, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 2], [[0], [0, 2]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0027/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 28. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2], [3, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 2], [[0], [0, 1], [0, 2], [0, 1, 2]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0028/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 29. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3], [1, 2, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 3], [[0], [0, 3]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0029/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 30. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3], [2, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 3], [[0], [0, 1], [0, 3], [0, 1, 3]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0030/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 31. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3], [1, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 2, 3], [[0], [0, 2], [0, 3], [0, 2, 3]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0031/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 32. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3], [4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 2, 3], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0032/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 33. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4], [1, 2, 3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 4], [[0], [0, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0033/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 34. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4], [2, 3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 4], [[0], [0, 1], [0, 4], [0, 1, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0034/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 35. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 4], [1, 3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 2, 4], [[0], [0, 2], [0, 4], [0, 2, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0035/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 36. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4], [3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 2, 4], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0036/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 37. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 4], [1, 2, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 3, 4], [[0], [0, 3], [0, 4], [0, 3, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0037/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 38. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4], [2, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 3, 4], [[0], [0, 1], [0, 3], [0, 1, 3], [0, 4], [0, 1, 4], [0, 3, 4], [0, 1, 3, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0038/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 39. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4], [1, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 2, 3, 4], [[0], [0, 2], [0, 3], [0, 2, 3], [0, 4], [0, 2, 4], [0, 3, 4], [0, 2, 3, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0039/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 40. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4], [5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

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

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 4], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4], [0, 3, 4], [0, 1, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0040/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 41. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 5], [1, 2, 3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

View answer and explanation(0, [0, 5], [[0], [0, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 5], [[0], [0, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0041/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 42. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 5], [2, 3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

View answer and explanation(0, [0, 1, 5], [[0], [0, 1], [0, 5], [0, 1, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 5], [[0], [0, 1], [0, 5], [0, 1, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0042/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 43. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 5], [1, 3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

View answer and explanation(0, [0, 2, 5], [[0], [0, 2], [0, 5], [0, 2, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 2, 5], [[0], [0, 2], [0, 5], [0, 2, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0043/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 44. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5], [3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

View answer and explanation(0, [0, 1, 2, 5], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 2, 5], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0044/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 45. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 5], [1, 2, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

View answer and explanation(0, [0, 3, 5], [[0], [0, 3], [0, 5], [0, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 3, 5], [[0], [0, 3], [0, 5], [0, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0045/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 46. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5], [2, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

View answer and explanation(0, [0, 1, 3, 5], [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 3, 5], [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0046/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 47. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5], [1, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

View answer and explanation(0, [0, 2, 3, 5], [[0], [0, 2], [0, 3], [0, 2, 3], [0, 5], [0, 2, 5], [0, 3, 5], [0, 2, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 2, 3, 5], [[0], [0, 2], [0, 3], [0, 2, 3], [0, 5], [0, 2, 5], [0, 3, 5], [0, 2, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0047/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 48. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5], [4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

View answer and explanation(0, [0, 1, 2, 3, 5], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5], [0, 3, 5], [0, 1, 3, 5], [0, 2, 3, 5], [0, 1, 2, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 5], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5], [0, 3, 5], [0, 1, 3, 5], [0, 2, 3, 5], [0, 1, 2, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0048/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 49. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4, 5], [1, 2, 3]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

View answer and explanation(0, [0, 4, 5], [[0], [0, 4], [0, 5], [0, 4, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 4, 5], [[0], [0, 4], [0, 5], [0, 4, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0049/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 50. The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5], [2, 3]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).

View answer and explanation(0, [0, 1, 4, 5], [[0], [0, 1], [0, 4], [0, 1, 4], [0, 5], [0, 1, 5], [0, 4, 5], [0, 1, 4, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)

Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 4, 5], [[0], [0, 1], [0, 4], [0, 1, 4], [0, 5], [0, 1, 5], [0, 4, 5], [0, 1, 4, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0050/)

Connected component of a topological space · qf-mod48-component-maximality · candidate





## 51. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1], [2, 3, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1], [2, 3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1], [2, 3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0051/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 52. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2], [1, 3, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 2], [1, 3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 2], [1, 3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0052/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 53. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2], [3, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1, 2], [3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 2], [3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0053/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 54. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3], [1, 2, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 3], [1, 2, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 3], [1, 2, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0054/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 55. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3], [2, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1, 3], [2, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 3], [2, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0055/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 56. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3], [1, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 2, 3], [1, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 2, 3], [1, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0056/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 57. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3], [4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1, 2, 3], [4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 2, 3], [4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0057/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 58. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4], [1, 2, 3, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 4], [1, 2, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 4], [1, 2, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0058/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 59. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4], [2, 3, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1, 4], [2, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 4], [2, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0059/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 60. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 4], [1, 3, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 2, 4], [1, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 2, 4], [1, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0060/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 61. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4], [3, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1, 2, 4], [3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 2, 4], [3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0061/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 62. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 4], [1, 2, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 3, 4], [1, 2, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 3, 4], [1, 2, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0062/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 63. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4], [2, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1, 3, 4], [2, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 3, 4], [2, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0063/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 64. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4], [1, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 2, 3, 4], [1, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 2, 3, 4], [1, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0064/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 65. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4], [5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1, 2, 3, 4], [5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4], [5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0065/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 66. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 5], [1, 2, 3, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 5], [1, 2, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 5], [1, 2, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0066/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 67. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 5], [2, 3, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1, 5], [2, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 5], [2, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0067/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 68. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 5], [1, 3, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 2, 5], [1, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 2, 5], [1, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0068/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 69. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5], [3, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1, 2, 5], [3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 2, 5], [3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0069/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 70. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 5], [1, 2, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 3, 5], [1, 2, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 3, 5], [1, 2, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0070/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 71. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5], [2, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1, 3, 5], [2, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 3, 5], [2, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0071/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 72. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5], [1, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 2, 3, 5], [1, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 2, 3, 5], [1, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0072/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 73. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5], [4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1, 2, 3, 5], [4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 5], [4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0073/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 74. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4, 5], [1, 2, 3]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 4, 5], [1, 2, 3]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 4, 5], [1, 2, 3]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0074/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 75. Finite Topological Components Partition the Space and Are Clopen

Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5], [2, 3]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).

View answer and explanation([[0, 1, 4, 5], [2, 3]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 4, 5], [2, 3]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0075/)

Connected component of a topological space · qf-mod48-component-partition · candidate





## 76. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_clopen and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_clopen, [[0, 1], [2, 3, 4, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1], [2, 3, 4, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0076/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 77. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_no_nonopen_witness and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}. Give (mode, components, all_clopen, general_nonopen_witness_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_no_nonopen_witness, [[0, 2], [1, 3, 4, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_no_nonopen_witness, [[0, 2], [1, 3, 4, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0077/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 78. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=graph_confusion and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]],"graph_edges":[[0,1],[1,2],[2,3],[3,4],[4,5]]}. Give (mode, topological_components, graph_components, different_partitions, structure_identification_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(graph_confusion, [[0, 1, 2], [3, 4, 5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (graph_confusion, [[0, 1, 2], [3, 4, 5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0078/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 79. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=cardinality_not_definition and complete model={"points":[0,1,2],"topology":[[],[0,1,2],[0],[1,2]],"basepoint":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(cardinality_not_definition, 0, [0], [1, 2], 1, 2, No, component_is_inclusion_maximal_not_global_maximum_cardinality)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (cardinality_not_definition, 0, [0], [1, 2], 1, 2, No, component_is_inclusion_maximal_not_global_maximum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0079/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 80. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_clopen and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_clopen, [[0, 1, 3], [2, 4, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1, 3], [2, 4, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0080/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 81. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_no_nonopen_witness and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}. Give (mode, components, all_clopen, general_nonopen_witness_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_no_nonopen_witness, [[0, 2, 3], [1, 4, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_no_nonopen_witness, [[0, 2, 3], [1, 4, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0081/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 82. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=graph_confusion and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]],"graph_edges":[[0,1],[1,2],[2,3],[3,4],[4,5]]}. Give (mode, topological_components, graph_components, different_partitions, structure_identification_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(graph_confusion, [[0, 1, 2, 3], [4, 5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (graph_confusion, [[0, 1, 2, 3], [4, 5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0082/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 83. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=cardinality_not_definition and complete model={"points":[0,1,2,3],"topology":[[],[0,1,2,3],[0],[1,2,3]],"basepoint":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(cardinality_not_definition, 0, [0], [1, 2, 3], 1, 3, No, component_is_inclusion_maximal_not_global_maximum_cardinality)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (cardinality_not_definition, 0, [0], [1, 2, 3], 1, 3, No, component_is_inclusion_maximal_not_global_maximum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0083/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 84. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_clopen and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_clopen, [[0, 1, 4], [2, 3, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1, 4], [2, 3, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0084/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 85. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_no_nonopen_witness and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}. Give (mode, components, all_clopen, general_nonopen_witness_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_no_nonopen_witness, [[0, 2, 4], [1, 3, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_no_nonopen_witness, [[0, 2, 4], [1, 3, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0085/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 86. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=graph_confusion and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]],"graph_edges":[[0,1],[1,2],[2,3],[3,4],[4,5]]}. Give (mode, topological_components, graph_components, different_partitions, structure_identification_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(graph_confusion, [[0, 1, 2, 4], [3, 5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (graph_confusion, [[0, 1, 2, 4], [3, 5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0086/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 87. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=cardinality_not_definition and complete model={"points":[0,1,2,3,4],"topology":[[],[0,1,2,3,4],[0],[1,2,3,4]],"basepoint":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(cardinality_not_definition, 0, [0], [1, 2, 3, 4], 1, 4, No, component_is_inclusion_maximal_not_global_maximum_cardinality)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (cardinality_not_definition, 0, [0], [1, 2, 3, 4], 1, 4, No, component_is_inclusion_maximal_not_global_maximum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0087/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 88. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_clopen and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_clopen, [[0, 1, 3, 4], [2, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1, 3, 4], [2, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0088/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 89. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_no_nonopen_witness and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]}. Give (mode, components, all_clopen, general_nonopen_witness_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_no_nonopen_witness, [[0, 2, 3, 4], [1, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_no_nonopen_witness, [[0, 2, 3, 4], [1, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0089/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 90. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=graph_confusion and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]],"graph_edges":[[0,1],[1,2],[2,3],[3,4],[4,5]]}. Give (mode, topological_components, graph_components, different_partitions, structure_identification_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(graph_confusion, [[0, 1, 2, 3, 4], [5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (graph_confusion, [[0, 1, 2, 3, 4], [5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0090/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 91. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=cardinality_not_definition and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0],[1,2,3,4,5]],"basepoint":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5], 1, 5, No, component_is_inclusion_maximal_not_global_maximum_cardinality)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5], 1, 5, No, component_is_inclusion_maximal_not_global_maximum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0091/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 92. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_clopen and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_clopen, [[0, 1, 5], [2, 3, 4]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1, 5], [2, 3, 4]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0092/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 93. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_no_nonopen_witness and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]}. Give (mode, components, all_clopen, general_nonopen_witness_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_no_nonopen_witness, [[0, 2, 5], [1, 3, 4]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_no_nonopen_witness, [[0, 2, 5], [1, 3, 4]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0093/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 94. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=graph_confusion and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]],"graph_edges":[[0,1],[1,2],[2,3],[3,4],[4,5]]}. Give (mode, topological_components, graph_components, different_partitions, structure_identification_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(graph_confusion, [[0, 1, 2, 5], [3, 4]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (graph_confusion, [[0, 1, 2, 5], [3, 4]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0094/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 95. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=cardinality_not_definition and complete model={"points":[0,1,2,3,4,5,6],"topology":[[],[0,1,2,3,4,5,6],[0],[1,2,3,4,5,6]],"basepoint":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6], 1, 6, No, component_is_inclusion_maximal_not_global_maximum_cardinality)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6], 1, 6, No, component_is_inclusion_maximal_not_global_maximum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0095/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 96. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_clopen and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_clopen, [[0, 1, 3, 5], [2, 4]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1, 3, 5], [2, 4]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0096/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 97. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_no_nonopen_witness and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}. Give (mode, components, all_clopen, general_nonopen_witness_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_no_nonopen_witness, [[0, 2, 3, 5], [1, 4]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_no_nonopen_witness, [[0, 2, 3, 5], [1, 4]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0097/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 98. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=graph_confusion and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]],"graph_edges":[[0,1],[1,2],[2,3],[3,4],[4,5]]}. Give (mode, topological_components, graph_components, different_partitions, structure_identification_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(graph_confusion, [[0, 1, 2, 3, 5], [4]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (graph_confusion, [[0, 1, 2, 3, 5], [4]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0098/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 99. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=cardinality_not_definition and complete model={"points":[0,1,2,3,4,5,6,7],"topology":[[],[0,1,2,3,4,5,6,7],[0],[1,2,3,4,5,6,7]],"basepoint":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6, 7], 1, 7, No, component_is_inclusion_maximal_not_global_maximum_cardinality)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6, 7], 1, 7, No, component_is_inclusion_maximal_not_global_maximum_cardinality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0099/)

Connected component of a topological space · qf-mod48-component-boundary · candidate





## 100. Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Use mode=finite_clopen and complete model={"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).

The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality.

View answer and explanation(finite_clopen, [[0, 1, 4, 5], [2, 3]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)

Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1, 4, 5], [2, 3]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r07-topological-connected-component-0100/)

Connected component of a topological space · qf-mod48-component-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261117-r07-topological-connected-component-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261117-r07-topological-connected-component-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261117-r07-topological-connected-component-foundations/validation-report.json)