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R0017-09276 · q-20261013-advance-20261018-r01-topological-space-0026
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 1], [0, 1, 2], [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. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
G001 — Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 1], [0, 1, 2], [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. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.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=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]
After
universe=[0, 1, 2, 3];canonical opens=[[], [0], [0, 1], [0, 1, 2], [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;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, S001
Before
universe Fin(4);open=[]
After
relative complement=[0, 1, 2, 3];closed=Yes
Conditions: none
A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open.
Rule: closed_as_relative_complement_of_open · Uses: G001, S002
Before
universe Fin(4);open=[0]
After
relative complement=[1, 2, 3];closed=Yes
Conditions: none
A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open.
Rule: closed_as_relative_complement_of_open · Uses: G001, S003
Before
universe Fin(4);open=[0, 1]
After
relative complement=[2, 3];closed=Yes
Conditions: none
A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open.
Rule: closed_as_relative_complement_of_open · Uses: G001, S004
Before
universe Fin(4);open=[0, 1, 2]
After
relative complement=[3];closed=Yes
Conditions: none
A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open.
Rule: closed_as_relative_complement_of_open · Uses: G001, S005
Before
universe Fin(4);open=[0, 1, 2, 3]
After
relative complement=[];closed=Yes
Conditions: none
A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open.
Rule: closed_as_relative_complement_of_open · Uses: G001, S006
Before
opens=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]];closed sets=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]
After
clopen sets=[[], [0, 1, 2, 3]]
Conditions: none
Exactly the members of both families are clopen.Empty and whole sets are included,and other sets can also be clopen.
Rule: clopen_intersection_of_properties · Uses: G001, S007
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, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
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: 52205abe4a19a2803c6d16b0925ab3533948c3b7a857aa8b86afe1043bab8a86candidate record SHA-256: 0db627042b9f08443329dbf8510e3edd33fcdff2428db739734fcffaa6dd981asemantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED