# Question 26535: The Union of All Connected Subsets through a Basepoint Is Its Component [Return to catalogue page 133](https://ocme.evemisslab.com/en/questions/page/133/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261117-r07-topological-connected-component-foundations/#q-20261015-advance-20261117-r07-topological-connected-component-0010) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0052-26535 · q-20261015-advance-20261117-r07-topological-connected-component-0010 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-topological-connected-component](https://ocme.evemisslab.com/en/mko/mko-topological-connected-component/) ## Step-by-step derivation S001 · impliesBefore source roles = [component-connected-subsets] After derived = [[basepoint, containing_connected_subsets, combined, connected, certificate]] Conditions: none List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [component ORIGINAL space, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 2, 4], [1, 3, 5]], No] After derived = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 2, 4], [1, 3, 5]], [[0, 0, 0, 0, Yes, Yes], [0, 1, 63, 0, Yes, Yes], [0, 2, 21, 0, Yes, Yes], [0, 3, 42, 0, Yes, Yes], [1, 0, 63, 0, Yes, Yes], [1, 1, 63, 63, Yes, Yes], [1, 2, 63, 21, Yes, Yes], [1, 3, 63, 42, Yes, Yes], [2, 0, 21, 0, Yes, Yes], [2, 1, 63, 21, Yes, Yes], [2, 2, 21, 21, Yes, Yes], [2, 3, 63, 0, Yes, Yes], [3, 0, 42, 0, Yes, Yes], [3, 1, 63, 42, Yes, Yes], [3, 2, 63, 0, Yes, Yes], [3, 3, 42, 42, Yes, Yes]]] Conditions: none Retain JSON types and English source ordering of sets and open families.Check union/intersection for every open pair and empty/full sets;in a finite family binary closure gives arbitrary-union closure,not sampled qualification. Step identifiers and dependenciesRule: Topology_full_typed_carrier_and_ALL_pair_axioms · Uses: G001, S001 S003 · impliesBefore source roles = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 2, 4], [1, 3, 5]]] After derived = [[[1, [0], [[], [0]], null, Yes], [2, [1], [[], [1]], null, Yes], [3, [0, 1], [[], [0, 1], [0], [1]], [0], No], [4, [2], [[], [2]], null, Yes], [5, [0, 2], [[], [0, 2]], null, Yes], [6, [1, 2], [[], [1, 2], [1], [2]], [1], No], [7, [0, 1, 2], [[], [0, 1, 2], [0, 2], [1]], [1], No], [8, [3], [[], [3]], null, Yes], [9, [0, 3], [[], [0, 3], [0], [3]], [0], No], [10, [1, 3], [[], [1, 3]], null, Yes], [11, [0, 1, 3], [[], [0, 1, 3], [0], [1, 3]], [0], No], [12, [2, 3], [[], [2, 3], [2], [3]], [2], No], [13, [0, 2, 3], [[], [0, 2, 3], [0, 2], [3]], [0, 2], No], [14, [1, 2, 3], [[], [1, 2, 3], [1, 3], [2]], [2], No], [15, [0, 1, 2, 3], [[], [0, 1, 2, 3], [0, 2], [1, 3]], [0, 2], No], [16, [4], [[], [4]], null, Yes], [17, [0, 4], [[], [0, 4]], null, Yes], [18, [1, 4], [[], [1, 4], [1], [4]], [1], No], [19, [0, 1, 4], [[], [0, 1, 4], [0, 4], [1]], [1], No], [20, [2, 4], [[], [2, 4]], null, Yes], [21, [0, 2, 4], [[], [0, 2, 4]], null, Yes], [22, [1, 2, 4], [[], [1, 2, 4], [1], [2, 4]], [1], No], [23, [0, 1, 2, 4], [[], [0, 1, 2, 4], [0, 2, 4], [1]], [1], No], [24, [3, 4], [[], [3, 4], [3], [4]], [3], No], [25, [0, 3, 4], [[], [0, 3, 4], [0, 4], [3]], [3], No], [26, [1, 3, 4], [[], [1, 3, 4], [1, 3], [4]], [1, 3], No], [27, [0, 1, 3, 4], [[], [0, 1, 3, 4], [0, 4], [1, 3]], [1, 3], No], [28, [2, 3, 4], [[], [2, 3, 4], [2, 4], [3]], [3], No], [29, [0, 2, 3, 4], [[], [0, 2, 3, 4], [0, 2, 4], [3]], [3], No], [30, [1, 2, 3, 4], [[], [1, 2, 3, 4], [1, 3], [2, 4]], [1, 3], No], [31, [0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0, 2, 4], [1, 3]], [1, 3], No], [32, [5], [[], [5]], null, Yes], [33, [0, 5], [[], [0, 5], [0], [5]], [0], No], [34, [1, 5], [[], [1, 5]], null, Yes], [35, [0, 1, 5], [[], [0, 1, 5], [0], [1, 5]], [0], No], [36, [2, 5], [[], [2, 5], [2], [5]], [2], No], [37, [0, 2, 5], [[], [0, 2, 5], [0, 2], [5]], [0, 2], No], [38, [1, 2, 5], [[], [1, 2, 5], [1, 5], [2]], [2], No], [39, [0, 1, 2, 5], [[], [0, 1, 2, 5], [0, 2], [1, 5]], [0, 2], No], [40, [3, 5], [[], [3, 5]], null, Yes], [41, [0, 3, 5], [[], [0, 3, 5], [0], [3, 5]], [0], No], [42, [1, 3, 5], [[], [1, 3, 5]], null, Yes], [43, [0, 1, 3, 5], [[], [0, 1, 3, 5], [0], [1, 3, 5]], [0], No], [44, [2, 3, 5], [[], [2, 3, 5], [2], [3, 5]], [2], No], [45, [0, 2, 3, 5], [[], [0, 2, 3, 5], [0, 2], [3, 5]], [0, 2], No], [46, [1, 2, 3, 5], [[], [1, 2, 3, 5], [1, 3, 5], [2]], [2], No], [47, [0, 1, 2, 3, 5], [[], [0, 1, 2, 3, 5], [0, 2], [1, 3, 5]], [0, 2], No], [48, [4, 5], [[], [4, 5], [4], [5]], [4], No], [49, [0, 4, 5], [[], [0, 4, 5], [0, 4], [5]], [0, 4], No], [50, [1, 4, 5], [[], [1, 4, 5], [1, 5], [4]], [4], No], [51, [0, 1, 4, 5], [[], [0, 1, 4, 5], [0, 4], [1, 5]], [0, 4], No], [52, [2, 4, 5], [[], [2, 4, 5], [2, 4], [5]], [2, 4], No], [53, [0, 2, 4, 5], [[], [0, 2, 4, 5], [0, 2, 4], [5]], [0, 2, 4], No], [54, [1, 2, 4, 5], [[], [1, 2, 4, 5], [1, 5], [2, 4]], [2, 4], No], [55, [0, 1, 2, 4, 5], [[], [0, 1, 2, 4, 5], [0, 2, 4], [1, 5]], [0, 2, 4], No], [56, [3, 4, 5], [[], [3, 4, 5], [3, 5], [4]], [4], No], [57, [0, 3, 4, 5], [[], [0, 3, 4, 5], [0, 4], [3, 5]], [0, 4], No], [58, [1, 3, 4, 5], [[], [1, 3, 4, 5], [1, 3, 5], [4]], [4], No], [59, [0, 1, 3, 4, 5], [[], [0, 1, 3, 4, 5], [0, 4], [1, 3, 5]], [0, 4], No], [60, [2, 3, 4, 5], [[], [2, 3, 4, 5], [2, 4], [3, 5]], [2, 4], No], [61, [0, 2, 3, 4, 5], [[], [0, 2, 3, 4, 5], [0, 2, 4], [3, 5]], [0, 2, 4], No], [62, [1, 2, 3, 4, 5], [[], [1, 2, 3, 4, 5], [1, 3, 5], [2, 4]], [2, 4], No], [63, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 2, 4], [1, 3, 5]], [0, 2, 4], No]], [[0], [1], [2], [0, 2], [3], [1, 3], [4], [0, 4], [2, 4], [0, 2, 4], [5], [1, 5], [3, 5], [1, 3, 5]]] Conditions: none For every nonempty subset generate all subspace opens O intersect A,then test every proper nonempty part and its relative complement for an open separation.Absence qualifies connectedness;retain numeric-mask order,including singletons. Step identifiers and dependenciesRule: Component_ALL_nonempty_masks_induced_topology_no_clopen_separation · Uses: G001, S002 S004 · impliesBefore source roles = [[[0], [1], [2], [0, 2], [3], [1, 3], [4], [0, 4], [2, 4], [0, 2, 4], [5], [1, 5], [3, 5], [1, 3, 5]]] After derived = [[[0, 2, 4], [1, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes]] Conditions: none Select inclusion-maximal members of the complete connected family,not maximum cardinality.Intersecting components have connected union and maximality makes them equal;singletons give coverage.The closure of a connected set remains connected,so components are closed;finitely many closed components make every complement closed,hence each is open here. Step identifiers and dependenciesRule: Component_inclusion_MAXIMAL_partition_finite_clopen · Uses: G001, S003 S005 · impliesBefore source roles = [0, [[0], [0, 2], [0, 4], [0, 2, 4]]] After derived = [[0, 2, 4], [0, 2, 4], Yes, Yes] Conditions: none All listed connected sets share the basepoint.A separation of their union puts every connected set on the side containing that point,leaving the other side empty,a contradiction.The union is connected and contains all such sets,hence is the unique inclusion-maximal component. Step identifiers and dependenciesRule: Component_basepoint_common_point_union_AND_maximality · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems. - S004 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems. - S003 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems. - S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems. - S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems. ## Result (0, [[0], [0, 2], [0, 4], [0, 2, 4]], [0, 2, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: c5e28e0420f163387e559346f77e255f98054ab968b322d7682c1ea9b905faf5 - candidate record SHA-256: 8c18b7f5eee0d2f09a52ae12a8524b8c5ba6c56590241734122afa40afe5a5f5 - semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252 - question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261117-r07-topological-connected-component-0010.json)