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R0017-09444 · q-20261013-advance-20261019-r03-topological-continuity-0044
Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 1, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.
G001 — Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 1, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.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=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]
After
universe=[0, 1, 2, 3];canonical opens=[[], [2], [1, 2], [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=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]
After
universe=[0, 1, 2, 3];canonical opens=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 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, 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=[2, 1, 3, 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
map=f;images=[2, 1, 3, 0];target open=[]
After
full preimage=[];source open=Yes
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, S005
Before
map=f;images=[2, 1, 3, 0];target open=[0]
After
full preimage=[3];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
Before
map=f;images=[2, 1, 3, 0];target open=[1]
After
full preimage=[1];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, S007
Before
map=f;images=[2, 1, 3, 0];target open=[0, 1]
After
full preimage=[1, 3];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, S008
Before
map=f;images=[2, 1, 3, 0];target open=[2]
After
full preimage=[0];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, S009
Before
map=f;images=[2, 1, 3, 0];target open=[0, 2]
After
full preimage=[0, 3];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, S010
Before
map=f;images=[2, 1, 3, 0];target open=[1, 2]
After
full preimage=[0, 1];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, S011
Before
map=f;images=[2, 1, 3, 0];target open=[0, 1, 2]
After
full preimage=[0, 1, 3];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, S012
Before
map=f;images=[2, 1, 3, 0];target open=[3]
After
full preimage=[2];source open=Yes
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, S013
Before
map=f;images=[2, 1, 3, 0];target open=[0, 3]
After
full preimage=[2, 3];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, S014
Before
map=f;images=[2, 1, 3, 0];target open=[1, 3]
After
full preimage=[1, 2];source open=Yes
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, S015
Before
map=f;images=[2, 1, 3, 0];target open=[0, 1, 3]
After
full preimage=[1, 2, 3];source open=Yes
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, S016
Before
map=f;images=[2, 1, 3, 0];target open=[2, 3]
After
full preimage=[0, 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, S017
Before
map=f;images=[2, 1, 3, 0];target open=[0, 2, 3]
After
full preimage=[0, 2, 3];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, S018
Before
map=f;images=[2, 1, 3, 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, S019
Before
map=f;images=[2, 1, 3, 0];target open=[0, 1, 2, 3]
After
full preimage=[0, 1, 2, 3];source open=Yes
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, S020
Before
map=f;all target opens=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]];all preimage rows=[[[], [], Yes], [[0], [3], No], [[1], [1], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [2, 3], No], [[1, 3], [1, 2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 2], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]
After
continuous=No
Conditions: none
Continuity requires an open preimage for every target open.A single failure refutes it;complete finite data permits affirmation only when all rows pass.One successful row or point sample is insufficient.
Rule: continuity_all_target_opens · Uses: G001, S021
S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([2, 1, 3, 0], [[[], [], Yes], [[0], [3], No], [[1], [1], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [2, 3], No], [[1, 3], [1, 2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 2], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], 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: cdd4f9c6b2db2f2423dbf4933a533e778da0015836e67e57af359517e15a1ccacandidate record SHA-256: 400506985dd47d1cdc0c833455dcc962230c767a6fe2471a203701fcc8248bc6semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED