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R0017-09499 · q-20261013-advance-20261019-r03-topological-continuity-0024
Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.
G001 — Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.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=X;n=4;raw family=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]
After
universe=[0, 1, 2, 3];canonical opens=[[], [3], [2, 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=X;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=Y;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, S002
Before
space=Y;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
f:Fin(4)->Fin(4);whole images=[3, 2, 1, 0]
After
all original image entries typed;source labels retained
Conditions: none
Validate every cell of the complete total table even for a query about the empty preimage.Retain original source indices;do not discard invalid cells because a query does not use them.
Rule: whole_topological_total_map_guard · Uses: G001, S004
Before
declared target open=[1, 2, 3];target family=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]
After
target-open prerequisite satisfied;one row is not whole continuity
Conditions: none
The supplied set must actually be open in the target.Passing this single preimage test does not establish continuity of the whole map.
Rule: single_preimage_not_complete_continuity · Uses: G001, S005
Before
map=f single query;images=[3, 2, 1, 0];target open=[1, 2, 3]
After
full preimage=[0, 1, 2];source open=No
Conditions: none
Collect every original source label with f(x) in U,then test the complete set in the source open family.A set preimage is not an inverse function and requires neither injectivity nor surjectivity.
Rule: complete_open_preimage_row · Uses: G001, S006
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, 4, [1, 2, 3], [0, 1, 2], No)
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: d11f38f97c85c520467c5b26b2b607a83a9e3f51e3d220930aad5328108aaf79candidate record SHA-256: b88113b7c353bcc31b9ca4cef733a3a9658903e0c27826008f2a1d99b95852a7semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED