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Inclusion-Maximal Connected Subspaces and Finite Clopen Partitions
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Connected component of a topological space · qf-mod48-component-boundary · candidate
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