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Question 26870: 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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R0052-26870 · q-20261015-advance-20261117-r05-quotient-topology-0095

Original question and givens

Use mode=fiber_split and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","a","b","a","b","b"],"subset":[0]}. Give (mode, source_subset, image, saturation, saturated, full_preimage_claim, boundary).

  • G001 — Use mode=fiber_split and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","a","b","a","b","b"],"subset":[0]}. Give (mode, source_subset, image, saturation, saturated, full_preimage_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, subset, image, saturation, saturated, full_preimage_claim, 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]], No]

After

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

After

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

After

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

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], []], 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, S004

S006 · implies

Before

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

After

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

After

derived = [[a], [0, 1, 3], No, No, No]

Conditions: none

First take q(A),then its FULL preimage.Splitting a fiber strictly enlarges the saturation.A source open need not have open image;image openness tests saturation,not A itself.

Step identifiers and dependencies

Rule: Quotient_actual_subset_image_FULL_saturation_separate_openness · Uses: G001, S006

Reverse checks

  • 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

(fiber_split, [0], [a], [0, 1, 3], No, No, only_saturated_subsets_are_full_preimages)

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: 06be1569504bd6f98f6c77fcf8e543113d9f24f2b503714fa283fd61cc8ffd15
  • candidate record SHA-256: dcb62346da84b5f5f0a30f2838298fd94c11e6d2f490633d5739f815e66555ca
  • semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
  • question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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