Question 26842: The Final-Topology Criterion Compares h with Its Composite with q

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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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R0052-26842 · q-20261015-advance-20261117-r05-quotient-topology-0067

Original question and givens

Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], target carrier=[a, b] and aligned total q=[a, a, b, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).

G001 — Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], target carrier=[a, b] and aligned total q=[a, a, b, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).

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-quotient-topology

Step-by-step derivation

S001 · implies

Before

source roles = [quotient-final-continuity]

After

derived = [[h_continuous, composite_continuous, equivalence, h_rows, composite_rows, certificate]]

Conditions: none

List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [quotient ORIGINAL X, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], No]

After

derived = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], [[0, 0, 0, 0, Yes, Yes], [0, 1, 63, 0, Yes, Yes], [0, 2, 35, 0, Yes, Yes], [1, 0, 63, 0, Yes, Yes], [1, 1, 63, 63, Yes, Yes], [1, 2, 63, 35, Yes, Yes], [2, 0, 35, 0, Yes, Yes], [2, 1, 63, 35, Yes, Yes], [2, 2, 35, 35, Yes, Yes]]]

Conditions: none

Retain JSON types and English source ordering of sets and open families.Check union/intersection for every open pair and empty/full sets;in a finite family binary closure gives arbitrary-union closure,not sampled qualification.

Step identifiers and dependencies

Rule: Topology_full_typed_carrier_and_ALL_pair_axioms · Uses: G001, S001

S003 · implies

Before

source roles = [[0, 1, 2, 3, 4, 5], [a, b], [a, a, b, b, b, a]]

After

derived = [[[0, a], [1, a], [2, b], [3, b], [4, b], [5, a]], [[a, [0, 1, 5]], [b, [2, 3, 4]]], Yes]

Conditions: none

Bind q to the ORIGINAL X order before sorting and compute the full fiber of every typed Y point.Totality gives exactly one fiber per x;surjectivity additionally makes every fiber nonempty.

Step identifiers and dependencies

Rule: Quotient_ORIGINAL_domain_alignment_ALL_fibers · Uses: G001, S002

S004 · implies

Before

source roles = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], [a, b], [[0, a], [1, a], [2, b], [3, b], [4, b], [5, a]]]

After

derived = [[[0, [], [], Yes], [1, [a], [0, 1, 5], Yes], [2, [b], [2, 3, 4], No], [3, [a, b], [0, 1, 2, 3, 4, 5], Yes]], [[a, b], [a], []]]

Conditions: none

For every subset of Y use openness of its full X preimage as an IFF.Preimages preserve empty/full sets,arbitrary unions and finite intersections,hence define a topology.The IFF uniquely determines each subset status even without surjectivity.

Step identifiers and dependencies

Rule: Quotient_ALL_target_subsets_PREIMAGE_IFF_unique_final_topology · Uses: G001, S003

S005 · implies

Before

source roles = [quotient GENERATED final topology, [a, b], [[a, b], [a], []], No]

After

derived = [[a, b], [[a, b], [a], []], [[0, 0, 3, 3, Yes, Yes], [0, 1, 3, 1, Yes, Yes], [0, 2, 3, 0, Yes, Yes], [1, 0, 3, 1, Yes, Yes], [1, 1, 1, 1, Yes, Yes], [1, 2, 1, 0, Yes, Yes], [2, 0, 3, 0, Yes, Yes], [2, 1, 1, 0, Yes, Yes], [2, 2, 0, 0, Yes, Yes]]]

Conditions: none

Step identifiers and dependencies

Rule: Topology_full_typed_carrier_and_ALL_pair_axioms · Uses: G001, S004

S006 · implies

Before

source roles = [[[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], [[0, a], [1, a], [2, b], [3, b], [4, b], [5, a]]]

After

derived = [[[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], Yes]

Conditions: none

Audit A=q^-1(q(A)) for every source open in original canonical order.Only saturated opens are full unions of fibers.For surjective q,preimage is injective on Y subsets and gives the correspondence;not every source open is a quotient open.

Step identifiers and dependencies

Rule: Quotient_ALL_source_open_saturation_correspondence · Uses: G001, S005

S007 · implies

Before

source roles = [quotient ORIGINAL Z, [0, 1], [[], [1], [0, 1]], No]

After

derived = [[0, 1], [[], [0, 1], [1]], [[0, 0, 0, 0, Yes, Yes], [0, 1, 3, 0, Yes, Yes], [0, 2, 2, 0, Yes, Yes], [1, 0, 3, 0, Yes, Yes], [1, 1, 3, 3, Yes, Yes], [1, 2, 3, 2, Yes, Yes], [2, 0, 2, 0, Yes, Yes], [2, 1, 3, 2, Yes, Yes], [2, 2, 2, 2, Yes, Yes]]]

Conditions: none

Step identifiers and dependencies

Rule: Topology_full_typed_carrier_and_ALL_pair_axioms · Uses: G001, S006

S008 · implies

Before

source roles = [quotient ORIGINAL h, [a, b], [[a, b], [a], []], [0, 1], [[], [0, 1], [1]], [[a, 0], [b, 1]]]

After

derived = [[[[], [], Yes], [[0, 1], [a, b], Yes], [[1], [b], No]], No]

Conditions: none

Test full preimages of every target open,not just points,open images or samples.Align images with the ORIGINAL domain before sorting the rows.

Step identifiers and dependencies

Rule: Continuity_ALL_target_open_preimages · Uses: G001, S007

S009 · implies

Before

source roles = [quotient ACTUAL h-after-q, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], [0, 1], [[], [0, 1], [1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 1], [5, 0]]]

After

derived = [[[[], [], Yes], [[0, 1], [0, 1, 2, 3, 4, 5], Yes], [[1], [2, 3, 4], No]], No]

Conditions: none

Test full preimages of every target open,not just points,open images or samples.Align images with the ORIGINAL domain before sorting the rows.

Step identifiers and dependencies

Rule: Continuity_ALL_target_open_preimages · Uses: G001, S008

S010 · implies

Before

source roles = [[[0, a], [1, a], [2, b], [3, b], [4, b], [5, a]], [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 1], [5, 0]], [[a, b], [a], []]]

After

derived = [No, No, Yes]

Conditions: none

For each Z open W,(h after q)^-1(W)=q^-1(h^-1(W)).The final-topology IFF says the right side is X-open exactly when h^-1(W) is Y-open.Conjoin over all W;both maps can be noncontinuous while the equivalence is true.

Step identifiers and dependencies

Rule: Quotient_final_UNIVERSAL_IFF_not_bare_q_continuity · Uses: G001, S009

Reverse checks

S010 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
S009 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
S008 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
S007 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
S006 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
S005 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
S004 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
S003 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.

Result

(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)

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: 8ca1b04b0eb26498eb822cb971e24312cbe00b1b01ded43a016d2edbeacadeb3
candidate record SHA-256: 5ad22eee132c9abd9386636e12e19f8ab10352f8356e574a871fb0bf6d27004a
semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
Lean: NOT_RUN · human review: NOT_PERFORMED

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