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Question 26869: Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0052-26869 · q-20261015-advance-20261117-r05-quotient-topology-0094

Original question and givens

Use mode=coarser_target and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,5]],"y_points":["a","b"],"q_map":["a","a","a","b","b","a"],"candidate_topology":[[],["a","b"]]}. Give (mode, candidate_topology, final_topology, q_continuous, properly_coarser, final_topology_identification_claim, boundary).

  • G001 — Use mode=coarser_target and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,5]],"y_points":["a","b"],"q_map":["a","a","a","b","b","a"],"candidate_topology":[[],["a","b"]]}. Give (mode, candidate_topology, final_topology, q_continuous, properly_coarser, final_topology_identification_claim, boundary).

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.

Step-by-step derivation

S001 · implies

Before

source roles = [quotient-topology-boundary]

After

derived = [[mode, candidate_topology, final_topology, q_continuous, proper, final_identification, 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, 2, 5]], No]

After

derived = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5]], [[0, 0, 0, 0, Yes, Yes], [0, 1, 63, 0, Yes, Yes], [0, 2, 39, 0, Yes, Yes], [1, 0, 63, 0, Yes, Yes], [1, 1, 63, 63, Yes, Yes], [1, 2, 63, 39, Yes, Yes], [2, 0, 39, 0, Yes, Yes], [2, 1, 63, 39, Yes, Yes], [2, 2, 39, 39, 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, a, b, b, a]]

After

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

After

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

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, S004

S006 · implies

Before

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

After

derived = [[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 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 candidate coarser topology, [a, b], [[], [a, b]], No]

After

derived = [[a, b], [[a, b], []], [[0, 0, 3, 3, Yes, Yes], [0, 1, 3, 0, Yes, Yes], [1, 0, 3, 0, Yes, Yes], [1, 1, 0, 0, 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, S006

S008 · implies

Before

source roles = [quotient q into candidate, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5]], [a, b], [[a, b], []], [[0, a], [1, a], [2, a], [3, b], [4, b], [5, a]]]

After

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

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

After

derived = [Yes, Yes, No]

Conditions: none

q remains continuous into the strictly coarser topology,but at least one final open is omitted.Continuity alone is necessary,not enough to identify the final topology.

Step identifiers and dependencies

Rule: Quotient_continuity_ONLY_NOT_final_identification · Uses: G001, S008

Reverse checks

  • 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

(coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology)

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: bea3e1301a37c907bc66375efbd8269c19f1ecbb29dbb2335e9f1190efde1622
  • candidate record SHA-256: c8afd3ed73509c8d086e781ad7dca752723f6a0201ab4f64cb3dca8110eb68e0
  • semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
  • question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON