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R0017-09247 · q-20261013-advance-20261018-r01-topological-space-0072
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
G001 — Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.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.
Before
space=given;n=4;raw family=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]
After
universe=[0, 1, 2, 3];canonical opens=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]
Conditions: none
Type every candidate as a subset of the same Fin(n),deduplicate elements and members,and order by binary masks.The open family and underlying set are distinct data layers.
Rule: finite_topology_normalize_relative_universe · Uses: G001
Before
space=given;canonical family=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]
After
self pairs are idempotent;unordered distinct pairs cover all ordered pairs
Conditions: none
Self pairs are covered by A∪A=A and A∩A=A.Commutativity means all distinct unordered pairs cover every ordered pair;this is not sampling.
Rule: binary_set_operation_symmetry_coverage · Uses: G001, S001
Before
space=given;A=[];B=[3]
After
union=[3];in family=Yes;intersection=[];in family=Yes
Conditions: none
Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.
Rule: finite_topology_complete_binary_pair · Uses: G001, S002
Before
space=given;A=[];B=[1, 3]
After
union=[1, 3];in family=Yes;intersection=[];in family=Yes
Conditions: none
Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.
Rule: finite_topology_complete_binary_pair · Uses: G001, S003
Before
space=given;A=[];B=[1, 2, 3]
After
union=[1, 2, 3];in family=Yes;intersection=[];in family=Yes
Conditions: none
Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.
Rule: finite_topology_complete_binary_pair · Uses: G001, S004
Before
space=given;A=[];B=[0, 1, 2, 3]
After
union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes
Conditions: none
Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.
Rule: finite_topology_complete_binary_pair · Uses: G001, S005
Before
space=given;A=[3];B=[1, 3]
After
union=[1, 3];in family=Yes;intersection=[3];in family=Yes
Conditions: none
Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.
Rule: finite_topology_complete_binary_pair · Uses: G001, S006
Before
space=given;A=[3];B=[1, 2, 3]
After
union=[1, 2, 3];in family=Yes;intersection=[3];in family=Yes
Conditions: none
Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.
Rule: finite_topology_complete_binary_pair · Uses: G001, S007
Before
space=given;A=[3];B=[0, 1, 2, 3]
After
union=[0, 1, 2, 3];in family=Yes;intersection=[3];in family=Yes
Conditions: none
Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.
Rule: finite_topology_complete_binary_pair · Uses: G001, S008
Before
space=given;A=[1, 3];B=[1, 2, 3]
After
union=[1, 2, 3];in family=Yes;intersection=[1, 3];in family=Yes
Conditions: none
Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.
Rule: finite_topology_complete_binary_pair · Uses: G001, S009
Before
space=given;A=[1, 3];B=[0, 1, 2, 3]
After
union=[0, 1, 2, 3];in family=Yes;intersection=[1, 3];in family=Yes
Conditions: none
Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.
Rule: finite_topology_complete_binary_pair · Uses: G001, S010
Before
space=given;A=[1, 2, 3];B=[0, 1, 2, 3]
After
union=[0, 1, 2, 3];in family=Yes;intersection=[1, 2, 3];in family=Yes
Conditions: none
Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.
Rule: finite_topology_complete_binary_pair · Uses: G001, S011
Before
space=given;empty=Yes;full=Yes;union closed=Yes;intersection closed=Yes
After
topology=Yes;first union failure=none;first intersection failure=none
Conditions: none
All four conditions are required.On a finite underlying set there are finitely many distinct opens,so an arbitrary union reduces to a finite distinct-member union,covered by binary closure and the empty union.The empty intersection needs the whole set.Vacuous binary closure does not waive empty/whole-set membership.
Rule: finite_topology_axioms_joint_gate · Uses: G001, S012
Before
topology qualified=Yes;family size=5;powerset size=16
After
classification=other topology
Conditions: none
Qualify the topology first.All2^n subsets give the discrete topology;otherwise only empty and whole give the indiscrete topology.On the empty space both coincide,and the protocol classifies discrete first.
Rule: topology_classification_after_qualification · Uses: G001, S013
S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, other-topology)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: bbc6b2f25f2baed4b880e9565ba2d68eace17e592559063066304b2124c47f33candidate record SHA-256: 1900a436c61ce21c957106329aa2062069319d51b80de79336d940f8b2937640semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED