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Question 26851: 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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R0052-26851 · q-20261015-advance-20261117-r05-quotient-topology-0076

Original question and givens

Use mode=non_surjective and complete model={"x_points":[0,1,2,3],"x_topology":[[],[0],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]}. Give (mode, fibers, surjective, final_topology, final_topology_exists, quotient_map_claim, boundary).

  • G001 — Use mode=non_surjective and complete model={"x_points":[0,1,2,3],"x_topology":[[],[0],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]}. Give (mode, fibers, surjective, final_topology, final_topology_exists, quotient_map_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, fibers, surjective, final_topology, final_exists, quotient_map_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], [[], [0], [0, 1, 2, 3]], No]

After

derived = [[0, 1, 2, 3], [[], [0, 1, 2, 3], [0]], [[0, 0, 0, 0, Yes, Yes], [0, 1, 15, 0, Yes, Yes], [0, 2, 1, 0, Yes, Yes], [1, 0, 15, 0, Yes, Yes], [1, 1, 15, 15, Yes, Yes], [1, 2, 15, 1, Yes, Yes], [2, 0, 1, 0, Yes, Yes], [2, 1, 15, 1, Yes, Yes], [2, 2, 1, 1, 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], [y0, y1, y2, y3, y4], [y0, y1, y2, y3]]

After

derived = [[[0, y0], [1, y1], [2, y2], [3, y3]], [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No]

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], [[], [0, 1, 2, 3], [0]], [y0, y1, y2, y3, y4], [[0, y0], [1, y1], [2, y2], [3, y3]]]

After

derived = [[[0, [], [], Yes], [1, [y0], [0], Yes], [2, [y1], [1], No], [3, [y0, y1], [0, 1], No], [4, [y2], [2], No], [5, [y0, y2], [0, 2], No], [6, [y1, y2], [1, 2], No], [7, [y0, y1, y2], [0, 1, 2], No], [8, [y3], [3], No], [9, [y0, y3], [0, 3], No], [10, [y1, y3], [1, 3], No], [11, [y0, y1, y3], [0, 1, 3], No], [12, [y2, y3], [2, 3], No], [13, [y0, y2, y3], [0, 2, 3], No], [14, [y1, y2, y3], [1, 2, 3], No], [15, [y0, y1, y2, y3], [0, 1, 2, 3], Yes], [16, [y4], [], Yes], [17, [y0, y4], [0], Yes], [18, [y1, y4], [1], No], [19, [y0, y1, y4], [0, 1], No], [20, [y2, y4], [2], No], [21, [y0, y2, y4], [0, 2], No], [22, [y1, y2, y4], [1, 2], No], [23, [y0, y1, y2, y4], [0, 1, 2], No], [24, [y3, y4], [3], No], [25, [y0, y3, y4], [0, 3], No], [26, [y1, y3, y4], [1, 3], No], [27, [y0, y1, y3, y4], [0, 1, 3], No], [28, [y2, y3, y4], [2, 3], No], [29, [y0, y2, y3, y4], [0, 2, 3], No], [30, [y1, y2, y3, y4], [1, 2, 3], No], [31, [y0, y1, y2, y3, y4], [0, 1, 2, 3], Yes]], [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []]]

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, [y0, y1, y2, y3, y4], [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []], No]

After

derived = [[y0, y1, y2, y3, y4], [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []], [[0, 0, 31, 31, Yes, Yes], [0, 1, 31, 15, Yes, Yes], [0, 2, 31, 17, Yes, Yes], [0, 3, 31, 1, Yes, Yes], [0, 4, 31, 16, Yes, Yes], [0, 5, 31, 0, Yes, Yes], [1, 0, 31, 15, Yes, Yes], [1, 1, 15, 15, Yes, Yes], [1, 2, 31, 1, Yes, Yes], [1, 3, 15, 1, Yes, Yes], [1, 4, 31, 0, Yes, Yes], [1, 5, 15, 0, Yes, Yes], [2, 0, 31, 17, Yes, Yes], [2, 1, 31, 1, Yes, Yes], [2, 2, 17, 17, Yes, Yes], [2, 3, 17, 1, Yes, Yes], [2, 4, 17, 16, Yes, Yes], [2, 5, 17, 0, Yes, Yes], [3, 0, 31, 1, Yes, Yes], [3, 1, 15, 1, Yes, Yes], [3, 2, 17, 1, Yes, Yes], [3, 3, 1, 1, Yes, Yes], [3, 4, 17, 0, Yes, Yes], [3, 5, 1, 0, Yes, Yes], [4, 0, 31, 16, Yes, Yes], [4, 1, 31, 0, Yes, Yes], [4, 2, 17, 16, Yes, Yes], [4, 3, 17, 0, Yes, Yes], [4, 4, 16, 16, Yes, Yes], [4, 5, 16, 0, Yes, Yes], [5, 0, 31, 0, Yes, Yes], [5, 1, 15, 0, Yes, Yes], [5, 2, 17, 0, Yes, Yes], [5, 3, 1, 0, Yes, Yes], [5, 4, 16, 0, Yes, Yes], [5, 5, 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], [0]], [[0, y0], [1, y1], [2, y2], [3, y3]]]

After

derived = [[[], [0, 1, 2, 3], [0]], No]

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, y0], [1, y1], [2, y2], [3, y3]], [y0, y1, y2, y3, y4], [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []]]

After

derived = [Yes, No]

Conditions: none

Missing fibers do not destroy existence or uniqueness of the preimage-IFF final topology.The quotient-map naming convention here requires surjectivity,so that naming claim fails.

Step identifiers and dependencies

Rule: Quotient_nonSURJ_unique_final_exists_but_naming_claim_NO · 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

(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)

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: e2d5b5676c93d3e295453a47978eb6b0eb657b35b073572bab5a44d8bdfcf2ec
  • candidate record SHA-256: bb1124cfaf4a235205893dcb2a782e1b3044d588d929dcbd819d13d683b77b81
  • semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
  • question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON