Question 9465: Open Images versus Open Preimages

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0017-09465 · q-20261013-advance-20261019-r03-topological-continuity-0090

Original question and givens

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 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, 0, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

G001 — Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 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, 0, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

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

Step-by-step derivation

S001 · implies

Before

space=X;n=4;raw family=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]

After

universe=[0, 1, 2, 3];canonical opens=[[], [2], [0, 2], [0, 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.

Step identifiers and dependencies

Rule: finite_topology_normalize_relative_universe · Uses: G001

S002 · implies

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.

Step identifiers and dependencies

Rule: finite_topology_axioms_joint_gate · Uses: G001, S001

S003 · implies

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.

Step identifiers and dependencies

Rule: finite_topology_normalize_relative_universe · Uses: G001, S002

S004 · implies

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

Step identifiers and dependencies

Rule: finite_topology_axioms_joint_gate · Uses: G001, S003

S005 · implies

Before

f:Fin(4)->Fin(4);whole images=[2, 0, 3, 1]

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 dependencies

Rule: whole_topological_total_map_guard · Uses: G001, S004

S006 · implies

Before

map=f;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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S005

S007 · implies

Before

map=f;images=[2, 0, 3, 1];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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S006

S008 · implies

Before

map=f;images=[2, 0, 3, 1];target open=[1]

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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S007

S009 · implies

Before

map=f;images=[2, 0, 3, 1];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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S008

S010 · implies

Before

map=f;images=[2, 0, 3, 1];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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S009

S011 · implies

Before

map=f;images=[2, 0, 3, 1];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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S010

S012 · implies

Before

map=f;images=[2, 0, 3, 1];target open=[1, 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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S011

S013 · implies

Before

map=f;images=[2, 0, 3, 1];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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S012

S014 · implies

Before

map=f;images=[2, 0, 3, 1];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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S013

S015 · implies

Before

map=f;images=[2, 0, 3, 1];target open=[0, 3]

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 dependencies

Rule: complete_open_preimage_row · Uses: G001, S014

S016 · implies

Before

map=f;images=[2, 0, 3, 1];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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S015

S017 · implies

Before

map=f;images=[2, 0, 3, 1];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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S016

S018 · implies

Before

map=f;images=[2, 0, 3, 1];target open=[2, 3]

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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S017

S019 · implies

Before

map=f;images=[2, 0, 3, 1];target open=[0, 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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S018

S020 · implies

Before

map=f;images=[2, 0, 3, 1];target open=[1, 2, 3]

After

full preimage=[0, 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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S019

S021 · implies

Before

map=f;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.

Step identifiers and dependencies

Rule: complete_open_preimage_row · Uses: G001, S020

S022 · implies

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], [3], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [1, 2], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 2], Yes], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 2, 3], Yes], [[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.

Step identifiers and dependencies

Rule: continuity_all_target_opens · Uses: G001, S021

S023 · implies

Before

source open=[];f images=[2, 0, 3, 1]

After

full direct image=[];target open=Yes

Conditions: none

An open map tests direct images of every source open set,the opposite direction from the continuity preimage test.Deduplicate image values without omitting source elements.

Step identifiers and dependencies

Rule: open_map_all_source_open_images · Uses: G001, S022

S024 · implies

Before

source open=[2];f images=[2, 0, 3, 1]

After

full direct image=[3];target open=Yes

Conditions: none

An open map tests direct images of every source open set,the opposite direction from the continuity preimage test.Deduplicate image values without omitting source elements.

Step identifiers and dependencies

Rule: open_map_all_source_open_images · Uses: G001, S023

S025 · implies

Before

source open=[0, 2];f images=[2, 0, 3, 1]

After

full direct image=[2, 3];target open=Yes

Conditions: none

An open map tests direct images of every source open set,the opposite direction from the continuity preimage test.Deduplicate image values without omitting source elements.

Step identifiers and dependencies

Rule: open_map_all_source_open_images · Uses: G001, S024

S026 · implies

Before

source open=[0, 2, 3];f images=[2, 0, 3, 1]

After

full direct image=[1, 2, 3];target open=Yes

Conditions: none

An open map tests direct images of every source open set,the opposite direction from the continuity preimage test.Deduplicate image values without omitting source elements.

Step identifiers and dependencies

Rule: open_map_all_source_open_images · Uses: G001, S025

S027 · implies

Before

source open=[0, 1, 2, 3];f images=[2, 0, 3, 1]

After

full direct image=[0, 1, 2, 3];target open=Yes

Conditions: none

An open map tests direct images of every source open set,the opposite direction from the continuity preimage test.Deduplicate image values without omitting source elements.

Step identifiers and dependencies

Rule: open_map_all_source_open_images · Uses: G001, S026

S028 · implies

Before

continuous=No;open map=Yes;image rows=[[[], [], Yes], [[2], [3], Yes], [[0, 2], [2, 3], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]

After

comparison=properties differ

Conditions: none

Even when the truth values agree in this instance,the definitions and quantified audit directions differ;do not identify the two properties.

Step identifiers and dependencies

Rule: continuity_and_open_map_distinct_obligations · Uses: G001, S027

Reverse checks

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.

Result

([2, 0, 3, 1], No, Yes, [[[], [], Yes], [[0], [1], No], [[1], [3], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [1, 2], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 2], Yes], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [3], Yes], [[0, 2], [2, 3], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

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: 404d1ceaef32eeb99985c5160c6b09649291463c6b927664d19e75439f5f3985
candidate record SHA-256: e13682bc7347bbf7c08eb1ad62f53c33fe6dc28bc7116d024900efca1616cad2
semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
Lean: NOT_RUN · human review: NOT_PERFORMED

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