# Question 17372: Exhaust the Complete Finite Topology before Certifying Connectedness [Return to catalogue page 87](https://ocme.evemisslab.com/en/questions/page/87/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261103-r05-connected-topological-space-foundations/#q-20261014-advance-20261103-r05-connected-topological-space-0072) 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. R0033-17372 · q-20261014-advance-20261103-r05-connected-topological-space-0072 ## Original question and givens Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists. - G001 — Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists. ## 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-connected-topological-space](https://ocme.evemisslab.com/en/mko/mko-connected-topological-space/) ## Step-by-step derivation S001 · impliesBefore ACTUAL carrier Fin(5);original ordered complete topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] After unique empty/whole opens present;empty carrier allowed;first union failure=null;first intersection failure=null Conditions: none The actual n may be0,when empty and whole are the same open rather than two distinct sets.Check canonical subsets and all binary union/intersection obligations;the finite family then handles arbitrary unions.This data fixes the actual space,not a reference label. Step identifiers and dependenciesRule: connected_actual_complete_topology_including_empty_carrier · Uses: G001 S002 · impliesBefore original open index=0;open=[];ACTUAL same-carrier complement=[0, 1, 2, 3, 4] After complement open=Yes;nonempty proper subset=No;nontrivial clopen obstruction=No Conditions: none The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness. Step identifiers and dependenciesRule: connected_every_actual_open_complement_not_selected_samples · Uses: G001, S001 S003 · impliesBefore original open index=1;open=[0];ACTUAL same-carrier complement=[1, 2, 3, 4] After complement open=No;nonempty proper subset=Yes;nontrivial clopen obstruction=No Conditions: none The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness. Step identifiers and dependenciesRule: connected_every_actual_open_complement_not_selected_samples · Uses: G001, S002 S004 · impliesBefore original open index=2;open=[0, 4];ACTUAL same-carrier complement=[1, 2, 3] After complement open=No;nonempty proper subset=Yes;nontrivial clopen obstruction=No Conditions: none The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness. Step identifiers and dependenciesRule: connected_every_actual_open_complement_not_selected_samples · Uses: G001, S003 S005 · impliesBefore original open index=3;open=[0, 2, 4];ACTUAL same-carrier complement=[1, 3] After complement open=No;nonempty proper subset=Yes;nontrivial clopen obstruction=No Conditions: none The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness. Step identifiers and dependenciesRule: connected_every_actual_open_complement_not_selected_samples · Uses: G001, S004 S006 · impliesBefore original open index=4;open=[0, 2, 3, 4];ACTUAL same-carrier complement=[1] After complement open=No;nonempty proper subset=Yes;nontrivial clopen obstruction=No Conditions: none The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness. Step identifiers and dependenciesRule: connected_every_actual_open_complement_not_selected_samples · Uses: G001, S005 S007 · impliesBefore original open index=5;open=[0, 1, 2, 3, 4];ACTUAL same-carrier complement=[] After complement open=Yes;nonempty proper subset=No;nontrivial clopen obstruction=No Conditions: none The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness. Step identifiers and dependenciesRule: connected_every_actual_open_complement_not_selected_samples · Uses: G001, S006 S008 · impliesBefore ALL actual clopen sets=[[], [0, 1, 2, 3, 4]];ALL nontrivial clopen sets=[] After a nontrivial clopen S gives separation(S,X minus S);a separation by disjoint nonempty opens gives such a clopen S;actual connected iff no such set Conditions: none A proper nonempty clopen and its open complement separate the carrier.Conversely the two opens in a separation are complements,so either is nontrivial clopen.Exhausting the complete finite topology therefore excludes every separation.This is not connectivity of an unrelated graph. Step identifiers and dependenciesRule: connected_clopen_separation_equivalence_full_finite_exhaustion · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_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: ae7be746d7bdacf3e4c16a7df7afc3f6e240206299aaf7aa07d76a0d05fc6c0a - candidate record SHA-256: 0ab3add6164b3eb15bc49624704deb6578d428327556b50f2b863db8fa6bc141 - semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3 - question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261103-r05-connected-topological-space-0072.json)