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R0017-09403 · q-20261013-advance-20261019-r03-topological-continuity-0053
Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 1, 2], [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, 0, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 0, 3, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.
G001 — Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 1, 2], [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, 0, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 0, 3, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.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], [0, 2], [0, 1, 2], [0, 1, 2, 3]]
After
universe=[0, 1, 2, 3];canonical opens=[[], [2], [0, 2], [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=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, 0, 1, 3]
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
space=Z;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, S005
Before
space=Z;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, S006
Before
g:Fin(4)->Fin(4);whole images=[2, 0, 3, 1]
After
all g entries typed before composing
Conditions: none
The second total table must have typed outputs for every original Y input,not only cells reached by f.
Rule: whole_second_map_before_composition · Uses: G001, S007
Before
original x=0;f(x)=2;g images=[2, 0, 3, 1]
After
g(f(x))=3
Conditions: none
Use f to obtain a Y label,then read the complete typed g table.The composition is g after f.
Rule: topological_composition_source_lookup · Uses: G001, S008
Before
original x=1;f(x)=0;g images=[2, 0, 3, 1]
After
g(f(x))=2
Conditions: none
Use f to obtain a Y label,then read the complete typed g table.The composition is g after f.
Rule: topological_composition_source_lookup · Uses: G001, S009
Before
original x=2;f(x)=1;g images=[2, 0, 3, 1]
After
g(f(x))=0
Conditions: none
Use f to obtain a Y label,then read the complete typed g table.The composition is g after f.
Rule: topological_composition_source_lookup · Uses: G001, S010
Before
original x=3;f(x)=3;g images=[2, 0, 3, 1]
After
g(f(x))=1
Conditions: none
Use f to obtain a Y label,then read the complete typed g table.The composition is g after f.
Rule: topological_composition_source_lookup · Uses: G001, S011
Before
map=f;images=[2, 0, 1, 3];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, S012
Before
map=f;images=[2, 0, 1, 3];target open=[0]
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, S013
Before
map=f;images=[2, 0, 1, 3];target open=[1]
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, S014
Before
map=f;images=[2, 0, 1, 3];target open=[0, 1]
After
full preimage=[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, S015
Before
map=f;images=[2, 0, 1, 3];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, S016
Before
map=f;images=[2, 0, 1, 3];target open=[0, 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, S017
Before
map=f;images=[2, 0, 1, 3];target open=[1, 2]
After
full preimage=[0, 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, S018
Before
map=f;images=[2, 0, 1, 3];target open=[0, 1, 2]
After
full preimage=[0, 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, S019
Before
map=f;images=[2, 0, 1, 3];target open=[3]
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, S020
Before
map=f;images=[2, 0, 1, 3];target open=[0, 3]
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, S021
Before
map=f;images=[2, 0, 1, 3];target open=[1, 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, S022
Before
map=f;images=[2, 0, 1, 3];target open=[0, 1, 3]
After
full preimage=[1, 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, S023
Before
map=f;images=[2, 0, 1, 3];target open=[2, 3]
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, S024
Before
map=f;images=[2, 0, 1, 3];target open=[0, 2, 3]
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, S025
Before
map=f;images=[2, 0, 1, 3];target open=[1, 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, S026
Before
map=f;images=[2, 0, 1, 3];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, S027
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], [1], No], [[1], [2], Yes], [[0, 1], [1, 2], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [1, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2, 3], 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, S028
Before
map=g;images=[2, 0, 3, 1];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, S029
Before
map=g;images=[2, 0, 3, 1];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, S030
Before
map=g;all target opens=[[], [0, 1, 2, 3]];all preimage rows=[[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]
After
continuous=Yes
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, S031
Before
map=g after f;images=[3, 2, 0, 1];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, S032
Before
map=g after f;images=[3, 2, 0, 1];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, S033
Before
map=g after f;all target opens=[[], [0, 1, 2, 3]];all preimage rows=[[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]
After
continuous=Yes
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, S034
Before
Z open=[];g preimage=[];f preimage of that=[]
After
composition preimage=[];set identity holds=Yes
Conditions: none
(g∘f)⁻¹U=f⁻¹(g⁻¹U).If both factors are continuous this is a sufficient argument.If a factor is discontinuous,still decide h from its complete preimage table;failed premises do not force h discontinuous.
Rule: composition_inverse_image_identity · Uses: G001, S035
Before
Z open=[0, 1, 2, 3];g preimage=[0, 1, 2, 3];f preimage of that=[0, 1, 2, 3]
After
composition preimage=[0, 1, 2, 3];set identity holds=Yes
Conditions: none
(g∘f)⁻¹U=f⁻¹(g⁻¹U).If both factors are continuous this is a sufficient argument.If a factor is discontinuous,still decide h from its complete preimage table;failed premises do not force h discontinuous.
Rule: composition_inverse_image_identity · Uses: G001, S036
Before
f continuous=No;g continuous=Yes
After
independently audited composition continuous=Yes
Conditions: none
Retain all three judgments.The composition can be continuous even if an individual factor is not;this does not contradict the sufficient composition theorem.
Rule: composition_sufficient_not_necessary_premises · Uses: G001, S037
S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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, 0, 1, 3], [2, 0, 3, 1], [3, 2, 0, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])
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: 4ef92ef1eafbd12dc69de5485bcdba14ad5cf9b3af73247dec6c7cf2668f41b5candidate record SHA-256: d0ec5b6ea719495839bd50267563307a54f4ccbe0b386c767efa1d20957b5e21semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED