# Question 26853: Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails [Return to catalogue page 135](https://ocme.evemisslab.com/en/questions/page/135/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261117-r05-quotient-topology-foundations/#q-20261015-advance-20261117-r05-quotient-topology-0078) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0052-26853 · q-20261015-advance-20261117-r05-quotient-topology-0078 ## 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]],"y_points":["a","b"],"q_map":["a","a","a","b","b","b"],"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]],"y_points":["a","b"],"q_map":["a","a","a","b","b","b"],"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. - [mko-quotient-topology](https://ocme.evemisslab.com/en/mko/mko-quotient-topology/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [quotient ORIGINAL X, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], No] After derived = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], [[0, 0, 0, 0, Yes, Yes], [0, 1, 63, 0, Yes, Yes], [0, 2, 7, 0, Yes, Yes], [1, 0, 63, 0, Yes, Yes], [1, 1, 63, 63, Yes, Yes], [1, 2, 63, 7, Yes, Yes], [2, 0, 7, 0, Yes, Yes], [2, 1, 63, 7, Yes, Yes], [2, 2, 7, 7, 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 dependenciesRule: Topology_full_typed_carrier_and_ALL_pair_axioms · Uses: G001, S001 S003 · impliesBefore source roles = [[0, 1, 2, 3, 4, 5], [a, b], [a, a, a, b, b, b]] After derived = [[[0, a], [1, a], [2, a], [3, b], [4, b], [5, b]], [[a, [0, 1, 2]], [b, [3, 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 dependenciesRule: Quotient_ORIGINAL_domain_alignment_ALL_fibers · Uses: G001, S002 S004 · impliesBefore source roles = [[0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], [a, b], [[0, a], [1, a], [2, a], [3, b], [4, b], [5, b]]] After derived = [[[0, [], [], Yes], [1, [a], [0, 1, 2], Yes], [2, [b], [3, 4, 5], 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 dependenciesRule: Quotient_ALL_target_subsets_PREIMAGE_IFF_unique_final_topology · Uses: G001, S003 S005 · impliesBefore 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 dependenciesRule: Topology_full_typed_carrier_and_ALL_pair_axioms · Uses: G001, S004 S006 · impliesBefore source roles = [[[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], [[0, a], [1, a], [2, a], [3, b], [4, b], [5, b]]] After derived = [[[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], 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 dependenciesRule: Quotient_ALL_source_open_saturation_correspondence · Uses: G001, S005 S007 · impliesBefore 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 dependenciesRule: Topology_full_typed_carrier_and_ALL_pair_axioms · Uses: G001, S006 S008 · impliesBefore source roles = [quotient q into candidate, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], [a, b], [[a, b], []], [[0, a], [1, a], [2, a], [3, b], [4, b], [5, b]]] 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 dependenciesRule: Continuity_ALL_target_open_preimages · Uses: G001, S007 S009 · impliesBefore source roles = [[[a, b], []], [[a, b], [a], []], [[0, a], [1, a], [2, a], [3, b], [4, b], [5, b]]] 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 dependenciesRule: 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: 072dd6f6cf00039cc91ef1ce1a1939f852361745454eb708caa102809aba2f15 - candidate record SHA-256: bed6ecfeae80aa1174def3bc3cb056060271e94d7e99b26469bd6cc84cc8ce92 - semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252 - question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261117-r05-quotient-topology-0078.json)