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Question 26565: The Basepoint Component Is Uniquely Inclusion-Maximal and Closed

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0052-26565 · q-20261015-advance-20261117-r07-topological-connected-component-0040

Original question and givens

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).

  • G001 — 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).

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.

Step-by-step derivation

S001 · implies

Before

source roles = [component-maximality]

After

derived = [[basepoint, component, containing_connected_subsets, maximal, closed, 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 dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [component ORIGINAL space, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4], [5]], No]

After

derived = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4], [5]], [[0, 0, 0, 0, Yes, Yes], [0, 1, 63, 0, Yes, Yes], [0, 2, 31, 0, Yes, Yes], [0, 3, 32, 0, Yes, Yes], [1, 0, 63, 0, Yes, Yes], [1, 1, 63, 63, Yes, Yes], [1, 2, 63, 31, Yes, Yes], [1, 3, 63, 32, Yes, Yes], [2, 0, 31, 0, Yes, Yes], [2, 1, 63, 31, Yes, Yes], [2, 2, 31, 31, Yes, Yes], [2, 3, 63, 0, Yes, Yes], [3, 0, 32, 0, Yes, Yes], [3, 1, 63, 32, Yes, Yes], [3, 2, 63, 0, Yes, Yes], [3, 3, 32, 32, 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 dependencies

Rule: Topology_full_typed_carrier_and_ALL_pair_axioms · Uses: G001, S001

S003 · implies

Before

source roles = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4], [5]]]

After

derived = [[[1, [0], [[], [0]], null, Yes], [2, [1], [[], [1]], null, Yes], [3, [0, 1], [[], [0, 1]], null, Yes], [4, [2], [[], [2]], null, Yes], [5, [0, 2], [[], [0, 2]], null, Yes], [6, [1, 2], [[], [1, 2]], null, Yes], [7, [0, 1, 2], [[], [0, 1, 2]], null, Yes], [8, [3], [[], [3]], null, Yes], [9, [0, 3], [[], [0, 3]], null, Yes], [10, [1, 3], [[], [1, 3]], null, Yes], [11, [0, 1, 3], [[], [0, 1, 3]], null, Yes], [12, [2, 3], [[], [2, 3]], null, Yes], [13, [0, 2, 3], [[], [0, 2, 3]], null, Yes], [14, [1, 2, 3], [[], [1, 2, 3]], null, Yes], [15, [0, 1, 2, 3], [[], [0, 1, 2, 3]], null, Yes], [16, [4], [[], [4]], null, Yes], [17, [0, 4], [[], [0, 4]], null, Yes], [18, [1, 4], [[], [1, 4]], null, Yes], [19, [0, 1, 4], [[], [0, 1, 4]], null, Yes], [20, [2, 4], [[], [2, 4]], null, Yes], [21, [0, 2, 4], [[], [0, 2, 4]], null, Yes], [22, [1, 2, 4], [[], [1, 2, 4]], null, Yes], [23, [0, 1, 2, 4], [[], [0, 1, 2, 4]], null, Yes], [24, [3, 4], [[], [3, 4]], null, Yes], [25, [0, 3, 4], [[], [0, 3, 4]], null, Yes], [26, [1, 3, 4], [[], [1, 3, 4]], null, Yes], [27, [0, 1, 3, 4], [[], [0, 1, 3, 4]], null, Yes], [28, [2, 3, 4], [[], [2, 3, 4]], null, Yes], [29, [0, 2, 3, 4], [[], [0, 2, 3, 4]], null, Yes], [30, [1, 2, 3, 4], [[], [1, 2, 3, 4]], null, Yes], [31, [0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4]], null, Yes], [32, [5], [[], [5]], null, Yes], [33, [0, 5], [[], [0, 5], [0], [5]], [0], No], [34, [1, 5], [[], [1, 5], [1], [5]], [1], No], [35, [0, 1, 5], [[], [0, 1, 5], [0, 1], [5]], [0, 1], 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, 2], [5]], [1, 2], No], [39, [0, 1, 2, 5], [[], [0, 1, 2, 5], [0, 1, 2], [5]], [0, 1, 2], No], [40, [3, 5], [[], [3, 5], [3], [5]], [3], No], [41, [0, 3, 5], [[], [0, 3, 5], [0, 3], [5]], [0, 3], No], [42, [1, 3, 5], [[], [1, 3, 5], [1, 3], [5]], [1, 3], No], [43, [0, 1, 3, 5], [[], [0, 1, 3, 5], [0, 1, 3], [5]], [0, 1, 3], No], [44, [2, 3, 5], [[], [2, 3, 5], [2, 3], [5]], [2, 3], No], [45, [0, 2, 3, 5], [[], [0, 2, 3, 5], [0, 2, 3], [5]], [0, 2, 3], No], [46, [1, 2, 3, 5], [[], [1, 2, 3, 5], [1, 2, 3], [5]], [1, 2, 3], No], [47, [0, 1, 2, 3, 5], [[], [0, 1, 2, 3, 5], [0, 1, 2, 3], [5]], [0, 1, 2, 3], 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, 4], [5]], [1, 4], No], [51, [0, 1, 4, 5], [[], [0, 1, 4, 5], [0, 1, 4], [5]], [0, 1, 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, 2, 4], [5]], [1, 2, 4], No], [55, [0, 1, 2, 4, 5], [[], [0, 1, 2, 4, 5], [0, 1, 2, 4], [5]], [0, 1, 2, 4], No], [56, [3, 4, 5], [[], [3, 4, 5], [3, 4], [5]], [3, 4], No], [57, [0, 3, 4, 5], [[], [0, 3, 4, 5], [0, 3, 4], [5]], [0, 3, 4], No], [58, [1, 3, 4, 5], [[], [1, 3, 4, 5], [1, 3, 4], [5]], [1, 3, 4], No], [59, [0, 1, 3, 4, 5], [[], [0, 1, 3, 4, 5], [0, 1, 3, 4], [5]], [0, 1, 3, 4], No], [60, [2, 3, 4, 5], [[], [2, 3, 4, 5], [2, 3, 4], [5]], [2, 3, 4], No], [61, [0, 2, 3, 4, 5], [[], [0, 2, 3, 4, 5], [0, 2, 3, 4], [5]], [0, 2, 3, 4], No], [62, [1, 2, 3, 4, 5], [[], [1, 2, 3, 4, 5], [1, 2, 3, 4], [5]], [1, 2, 3, 4], No], [63, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4], [5]], [0, 1, 2, 3, 4], No]], [[0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3], [4], [0, 4], [1, 4], [0, 1, 4], [2, 4], [0, 2, 4], [1, 2, 4], [0, 1, 2, 4], [3, 4], [0, 3, 4], [1, 3, 4], [0, 1, 3, 4], [2, 3, 4], [0, 2, 3, 4], [1, 2, 3, 4], [0, 1, 2, 3, 4], [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 dependencies

Rule: Component_ALL_nonempty_masks_induced_topology_no_clopen_separation · Uses: G001, S002

S004 · implies

Before

source roles = [[[0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3], [4], [0, 4], [1, 4], [0, 1, 4], [2, 4], [0, 2, 4], [1, 2, 4], [0, 1, 2, 4], [3, 4], [0, 3, 4], [1, 3, 4], [0, 1, 3, 4], [2, 3, 4], [0, 2, 3, 4], [1, 2, 3, 4], [0, 1, 2, 3, 4], [5]]]

After

derived = [[[0, 1, 2, 3, 4], [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 dependencies

Rule: Component_inclusion_MAXIMAL_partition_finite_clopen · Uses: G001, S003

S005 · implies

Before

source roles = [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]]]

After

derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 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 dependencies

Rule: 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, 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)

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: 3fbfa3024a3e4c82289ec21cf6d9940fb625c37767035db5036a598385eb5a87
  • candidate record SHA-256: aa996a91647efb98a08aacd1296efeca74431861209c7d3b91d53c901d42a4c2
  • semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
  • question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON