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Question 26579: Finite Topological Components Partition the Space and Are Clopen

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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R0052-26579 · q-20261015-advance-20261117-r07-topological-connected-component-0054

Original question and givens

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).

  • G001 — 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).

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-partition]

After

derived = [[components, disjoint, cover, closed_flags, open_flags, valid, 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, 3], [1, 2, 4, 5]], No]

After

derived = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 3], [1, 2, 4, 5]], [[0, 0, 0, 0, Yes, Yes], [0, 1, 63, 0, Yes, Yes], [0, 2, 9, 0, Yes, Yes], [0, 3, 54, 0, Yes, Yes], [1, 0, 63, 0, Yes, Yes], [1, 1, 63, 63, Yes, Yes], [1, 2, 63, 9, Yes, Yes], [1, 3, 63, 54, Yes, Yes], [2, 0, 9, 0, Yes, Yes], [2, 1, 63, 9, Yes, Yes], [2, 2, 9, 9, Yes, Yes], [2, 3, 63, 0, Yes, Yes], [3, 0, 54, 0, Yes, Yes], [3, 1, 63, 54, Yes, Yes], [3, 2, 63, 0, Yes, Yes], [3, 3, 54, 54, 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, 3], [1, 2, 4, 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], [0], [2]], [0], No], [6, [1, 2], [[], [1, 2]], null, Yes], [7, [0, 1, 2], [[], [0, 1, 2], [0], [1, 2]], [0], No], [8, [3], [[], [3]], null, Yes], [9, [0, 3], [[], [0, 3]], null, Yes], [10, [1, 3], [[], [1, 3], [1], [3]], [1], No], [11, [0, 1, 3], [[], [0, 1, 3], [0, 3], [1]], [1], No], [12, [2, 3], [[], [2, 3], [2], [3]], [2], No], [13, [0, 2, 3], [[], [0, 2, 3], [0, 3], [2]], [2], No], [14, [1, 2, 3], [[], [1, 2, 3], [1, 2], [3]], [1, 2], No], [15, [0, 1, 2, 3], [[], [0, 1, 2, 3], [0, 3], [1, 2]], [1, 2], No], [16, [4], [[], [4]], null, Yes], [17, [0, 4], [[], [0, 4], [0], [4]], [0], No], [18, [1, 4], [[], [1, 4]], null, Yes], [19, [0, 1, 4], [[], [0, 1, 4], [0], [1, 4]], [0], No], [20, [2, 4], [[], [2, 4]], null, Yes], [21, [0, 2, 4], [[], [0, 2, 4], [0], [2, 4]], [0], No], [22, [1, 2, 4], [[], [1, 2, 4]], null, Yes], [23, [0, 1, 2, 4], [[], [0, 1, 2, 4], [0], [1, 2, 4]], [0], No], [24, [3, 4], [[], [3, 4], [3], [4]], [3], No], [25, [0, 3, 4], [[], [0, 3, 4], [0, 3], [4]], [0, 3], No], [26, [1, 3, 4], [[], [1, 3, 4], [1, 4], [3]], [3], No], [27, [0, 1, 3, 4], [[], [0, 1, 3, 4], [0, 3], [1, 4]], [0, 3], No], [28, [2, 3, 4], [[], [2, 3, 4], [2, 4], [3]], [3], No], [29, [0, 2, 3, 4], [[], [0, 2, 3, 4], [0, 3], [2, 4]], [0, 3], No], [30, [1, 2, 3, 4], [[], [1, 2, 3, 4], [1, 2, 4], [3]], [3], No], [31, [0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0, 3], [1, 2, 4]], [0, 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]], null, Yes], [37, [0, 2, 5], [[], [0, 2, 5], [0], [2, 5]], [0], No], [38, [1, 2, 5], [[], [1, 2, 5]], null, Yes], [39, [0, 1, 2, 5], [[], [0, 1, 2, 5], [0], [1, 2, 5]], [0], 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, 5], [3]], [3], No], [43, [0, 1, 3, 5], [[], [0, 1, 3, 5], [0, 3], [1, 5]], [0, 3], No], [44, [2, 3, 5], [[], [2, 3, 5], [2, 5], [3]], [3], No], [45, [0, 2, 3, 5], [[], [0, 2, 3, 5], [0, 3], [2, 5]], [0, 3], No], [46, [1, 2, 3, 5], [[], [1, 2, 3, 5], [1, 2, 5], [3]], [3], No], [47, [0, 1, 2, 3, 5], [[], [0, 1, 2, 3, 5], [0, 3], [1, 2, 5]], [0, 3], No], [48, [4, 5], [[], [4, 5]], null, Yes], [49, [0, 4, 5], [[], [0, 4, 5], [0], [4, 5]], [0], No], [50, [1, 4, 5], [[], [1, 4, 5]], null, Yes], [51, [0, 1, 4, 5], [[], [0, 1, 4, 5], [0], [1, 4, 5]], [0], No], [52, [2, 4, 5], [[], [2, 4, 5]], null, Yes], [53, [0, 2, 4, 5], [[], [0, 2, 4, 5], [0], [2, 4, 5]], [0], No], [54, [1, 2, 4, 5], [[], [1, 2, 4, 5]], null, Yes], [55, [0, 1, 2, 4, 5], [[], [0, 1, 2, 4, 5], [0], [1, 2, 4, 5]], [0], No], [56, [3, 4, 5], [[], [3, 4, 5], [3], [4, 5]], [3], No], [57, [0, 3, 4, 5], [[], [0, 3, 4, 5], [0, 3], [4, 5]], [0, 3], No], [58, [1, 3, 4, 5], [[], [1, 3, 4, 5], [1, 4, 5], [3]], [3], No], [59, [0, 1, 3, 4, 5], [[], [0, 1, 3, 4, 5], [0, 3], [1, 4, 5]], [0, 3], No], [60, [2, 3, 4, 5], [[], [2, 3, 4, 5], [2, 4, 5], [3]], [3], No], [61, [0, 2, 3, 4, 5], [[], [0, 2, 3, 4, 5], [0, 3], [2, 4, 5]], [0, 3], No], [62, [1, 2, 3, 4, 5], [[], [1, 2, 3, 4, 5], [1, 2, 4, 5], [3]], [3], No], [63, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 3], [1, 2, 4, 5]], [0, 3], No]], [[0], [1], [2], [1, 2], [3], [0, 3], [4], [1, 4], [2, 4], [1, 2, 4], [5], [1, 5], [2, 5], [1, 2, 5], [4, 5], [1, 4, 5], [2, 4, 5], [1, 2, 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], [2], [1, 2], [3], [0, 3], [4], [1, 4], [2, 4], [1, 2, 4], [5], [1, 5], [2, 5], [1, 2, 5], [4, 5], [1, 4, 5], [2, 4, 5], [1, 2, 4, 5]]]

After

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

Reverse checks

  • 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, 3], [1, 2, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)

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: 077c0b6289fc25b3e8661128302bfa85728c3efe8f271ae4e511403fab404447
  • candidate record SHA-256: b8aef52833dc65b886b72b17377b39a9526aeef0ec6327ebce227e007166d505
  • semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
  • question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON