# Question 9450: Every Target Open Set and Its Full Preimage

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0017-09450 · q-20261013-advance-20261019-r03-topological-continuity-0050



## Original question and givens

Use complete finite topologies X=Fin(3)={0,…,2}, opens_X=[[], [0], [0, 1], [0, 1, 2]] and Y=Fin(3)={0,…,2}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]. The complete total map has images=[0, 1, 2] 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(3)={0,…,2}, opens_X=[[], [0], [0, 1], [0, 1, 2]] and Y=Fin(3)={0,…,2}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]. The complete total map has images=[0, 1, 2] 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.





## Definition and scope

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.


- [mko-topological-continuity](https://ocme.evemisslab.com/en/mko/mko-topological-continuity/)





## Step-by-step derivation

S001 · impliesBefore

space=X;n=3;raw family=[[], [0], [0, 1], [0, 1, 2]]

After

universe=[0, 1, 2];canonical opens=[[], [0], [0, 1], [0, 1, 2]]

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.

Step identifiers and dependenciesRule: finite_topology_normalize_relative_universe · Uses: G001





S002 · impliesBefore

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.

Step identifiers and dependenciesRule: finite_topology_axioms_joint_gate · Uses: G001, S001





S003 · impliesBefore

space=Y;n=3;raw family=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]

After

universe=[0, 1, 2];canonical opens=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]

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.

Step identifiers and dependenciesRule: finite_topology_normalize_relative_universe · Uses: G001, S002





S004 · impliesBefore

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.

Step identifiers and dependenciesRule: finite_topology_axioms_joint_gate · Uses: G001, S003





S005 · impliesBefore

f:Fin(3)->Fin(3);whole images=[0, 1, 2]

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.

Step identifiers and dependenciesRule: whole_topological_total_map_guard · Uses: G001, S004





S006 · impliesBefore

map=f;images=[0, 1, 2];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.

Step identifiers and dependenciesRule: complete_open_preimage_row · Uses: G001, S005





S007 · impliesBefore

map=f;images=[0, 1, 2];target open=[0]

After

full preimage=[0];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.

Step identifiers and dependenciesRule: complete_open_preimage_row · Uses: G001, S006





S008 · impliesBefore

map=f;images=[0, 1, 2];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.

Step identifiers and dependenciesRule: complete_open_preimage_row · Uses: G001, S007





S009 · impliesBefore

map=f;images=[0, 1, 2];target open=[0, 1]

After

full preimage=[0, 1];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.

Step identifiers and dependenciesRule: complete_open_preimage_row · Uses: G001, S008





S010 · impliesBefore

map=f;images=[0, 1, 2];target open=[2]

After

full preimage=[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.

Step identifiers and dependenciesRule: complete_open_preimage_row · Uses: G001, S009





S011 · impliesBefore

map=f;images=[0, 1, 2];target open=[0, 2]

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.

Step identifiers and dependenciesRule: complete_open_preimage_row · Uses: G001, S010





S012 · impliesBefore

map=f;images=[0, 1, 2];target open=[1, 2]

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.

Step identifiers and dependenciesRule: complete_open_preimage_row · Uses: G001, S011





S013 · impliesBefore

map=f;images=[0, 1, 2];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.

Step identifiers and dependenciesRule: complete_open_preimage_row · Uses: G001, S012





S014 · impliesBefore

map=f;all target opens=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]];all preimage rows=[[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], 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.

Step identifiers and dependenciesRule: continuity_all_target_opens · Uses: G001, S013









## Reverse checks


- 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.





## Result

([0, 1, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes]], No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 67020d828b30f612ae5a1646bf4b1a939fcb98c3881a934534b3a05bd3caebce
- candidate record SHA-256: abb3bcca47067402ed6c4e1e7b1a3af6c33a869d8fa5c8add7244825e523f0c5
- semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
- question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261019-r03-topological-continuity-0050.json)