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R0017-09254 · q-20261013-advance-20261018-r01-topological-space-0079
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
G001 — Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.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=A;n=4;raw family=[[], [0, 1, 2, 3]]
After
universe=[0, 1, 2, 3];canonical opens=[[], [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=A;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
space=B;n=4;raw family=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]
After
universe=[0, 1, 2, 3];canonical opens=[[], [3], [0, 3], [0, 1, 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, S002
Before
space=B;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, S003
Before
direction=A in B;open member=[];other family=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]
After
member included=Yes
Conditions: none
Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.
Rule: topology_open_family_inclusion_member · Uses: G001, S004
Before
direction=A in B;open member=[0, 1, 2, 3];other family=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]
After
member included=Yes
Conditions: none
Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.
Rule: topology_open_family_inclusion_member · Uses: G001, S005
Before
direction=B in A;open member=[];other family=[[], [0, 1, 2, 3]]
After
member included=Yes
Conditions: none
Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.
Rule: topology_open_family_inclusion_member · Uses: G001, S006
Before
direction=B in A;open member=[3];other family=[[], [0, 1, 2, 3]]
After
member included=No
Conditions: none
Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.
Rule: topology_open_family_inclusion_member · Uses: G001, S007
Before
direction=B in A;open member=[0, 3];other family=[[], [0, 1, 2, 3]]
After
member included=No
Conditions: none
Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.
Rule: topology_open_family_inclusion_member · Uses: G001, S008
Before
direction=B in A;open member=[0, 1, 3];other family=[[], [0, 1, 2, 3]]
After
member included=No
Conditions: none
Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.
Rule: topology_open_family_inclusion_member · Uses: G001, S009
Before
direction=B in A;open member=[0, 1, 2, 3];other family=[[], [0, 1, 2, 3]]
After
member included=Yes
Conditions: none
Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.
Rule: topology_open_family_inclusion_member · Uses: G001, S010
Before
A subset B=Yes;B subset A=No
After
topology relation=A coarser
Conditions: none
If every A-open is B-open,B is finer and A coarser.Both inclusions mean equality;neither means incomparable.
Rule: finer_coarser_direction · Uses: G001, S011
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, [[], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
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: ac1726ae19d7bd9978ec789bffc144aa4b374bff1307ccb99c23c420ed374899candidate record SHA-256: e4b836544244610b17529d35596b65a0eff6d7b0344020576f60b5751d0413f2semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED