# Question 22911: All Distinct Point Pairs Must Be Separated in the Complete Finite Topology [Return to catalogue page 115](https://ocme.evemisslab.com/en/questions/page/115/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261111-r03-hausdorff-space-foundations/#q-20261014-advance-20261111-r03-hausdorff-space-0036) 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. R0044-22911 · q-20261014-advance-20261111-r03-hausdorff-space-0036 ## Original question and givens Use finite unique labeled points=[a10, b10, c10] and the complete canonical topology=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate). - G001 — Use finite unique labeled points=[a10, b10, c10] and the complete canonical topology=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate). ## 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-hausdorff-space](https://ocme.evemisslab.com/en/mko/mko-hausdorff-space/) ## Step-by-step derivation S001 · impliesBefore source operation=hausdorff-full-audit;original snapshot and mathematics retained After explicit ordered source output fields=[points, pair_witness_rows, Hausdorff, certificate] Conditions: none Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL carrier order=[a10, b10, c10];source canonical open family=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []];actual open index=0,U=[a10, b10, c10] After ALL [other_index,union,intersection] rows=[[0, [a10, b10, c10], [a10, b10, c10]], [1, [a10, b10, c10], [a10, b10]], [2, [a10, b10, c10], [a10, c10]], [3, [a10, b10, c10], [a10]], [4, [a10, b10, c10], [b10, c10]], [5, [a10, b10, c10], [b10]], [6, [a10, b10, c10], [c10]], [7, [a10, b10, c10], []]];empty/full present;family qualification passed with exact source English collation Conditions: none Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]. Step identifiers and dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S001 S003 · impliesBefore ORIGINAL carrier order=[a10, b10, c10];source canonical open family=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []];actual open index=1,U=[a10, b10] After ALL [other_index,union,intersection] rows=[[0, [a10, b10, c10], [a10, b10]], [1, [a10, b10], [a10, b10]], [2, [a10, b10, c10], [a10]], [3, [a10, b10], [a10]], [4, [a10, b10, c10], [b10]], [5, [a10, b10], [b10]], [6, [a10, b10, c10], []], [7, [a10, b10], []]];empty/full present;family qualification passed with exact source English collation Conditions: none Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]. Step identifiers and dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S002 S004 · impliesBefore ORIGINAL carrier order=[a10, b10, c10];source canonical open family=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []];actual open index=2,U=[a10, c10] After ALL [other_index,union,intersection] rows=[[0, [a10, b10, c10], [a10, c10]], [1, [a10, b10, c10], [a10]], [2, [a10, c10], [a10, c10]], [3, [a10, c10], [a10]], [4, [a10, b10, c10], [c10]], [5, [a10, b10, c10], []], [6, [a10, c10], [c10]], [7, [a10, c10], []]];empty/full present;family qualification passed with exact source English collation Conditions: none Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]. Step identifiers and dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S003 S005 · impliesBefore ORIGINAL carrier order=[a10, b10, c10];source canonical open family=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []];actual open index=3,U=[a10] After ALL [other_index,union,intersection] rows=[[0, [a10, b10, c10], [a10]], [1, [a10, b10], [a10]], [2, [a10, c10], [a10]], [3, [a10], [a10]], [4, [a10, b10, c10], []], [5, [a10, b10], []], [6, [a10, c10], []], [7, [a10], []]];empty/full present;family qualification passed with exact source English collation Conditions: none Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]. Step identifiers and dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S004 S006 · impliesBefore ORIGINAL carrier order=[a10, b10, c10];source canonical open family=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []];actual open index=4,U=[b10, c10] After ALL [other_index,union,intersection] rows=[[0, [a10, b10, c10], [b10, c10]], [1, [a10, b10, c10], [b10]], [2, [a10, b10, c10], [c10]], [3, [a10, b10, c10], []], [4, [b10, c10], [b10, c10]], [5, [b10, c10], [b10]], [6, [b10, c10], [c10]], [7, [b10, c10], []]];empty/full present;family qualification passed with exact source English collation Conditions: none Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]. Step identifiers and dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S005 S007 · impliesBefore ORIGINAL carrier order=[a10, b10, c10];source canonical open family=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []];actual open index=5,U=[b10] After ALL [other_index,union,intersection] rows=[[0, [a10, b10, c10], [b10]], [1, [a10, b10], [b10]], [2, [a10, b10, c10], []], [3, [a10, b10], []], [4, [b10, c10], [b10]], [5, [b10], [b10]], [6, [b10, c10], []], [7, [b10], []]];empty/full present;family qualification passed with exact source English collation Conditions: none Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]. Step identifiers and dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S006 S008 · impliesBefore ORIGINAL carrier order=[a10, b10, c10];source canonical open family=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []];actual open index=6,U=[c10] After ALL [other_index,union,intersection] rows=[[0, [a10, b10, c10], [c10]], [1, [a10, b10, c10], []], [2, [a10, c10], [c10]], [3, [a10, c10], []], [4, [b10, c10], [c10]], [5, [b10, c10], []], [6, [c10], [c10]], [7, [c10], []]];empty/full present;family qualification passed with exact source English collation Conditions: none Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]. Step identifiers and dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S007 S009 · impliesBefore ORIGINAL carrier order=[a10, b10, c10];source canonical open family=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []];actual open index=7,U=[] After ALL [other_index,union,intersection] rows=[[0, [a10, b10, c10], []], [1, [a10, b10], []], [2, [a10, c10], []], [3, [a10], []], [4, [b10, c10], []], [5, [b10], []], [6, [c10], []], [7, [], []]];empty/full present;family qualification passed with exact source English collation Conditions: none Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]. Step identifiers and dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S008 S010 · impliesBefore Hausdorff ORIGINAL pair index0:ORIGINAL distinct carrier points x=a10,y=b10;source open order=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []] After ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a10, b10, c10], No], [0, 1, [a10, b10], No], [0, 4, [b10, c10], No], [0, 5, [b10], No], [1, 0, [a10, b10], No], [1, 1, [a10, b10], No], [1, 4, [b10], No], [1, 5, [b10], No], [2, 0, [a10, c10], No], [2, 1, [a10], No], [2, 4, [c10], No], [2, 5, [], Yes]];FIRST source-order witness=[[a10, c10], [b10]] Conditions: none Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample. Step identifiers and dependenciesRule: hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure · Uses: G001, S009 S011 · impliesBefore Hausdorff ORIGINAL pair index1:ORIGINAL distinct carrier points x=a10,y=c10;source open order=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []] After ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a10, b10, c10], No], [0, 2, [a10, c10], No], [0, 4, [b10, c10], No], [0, 6, [c10], No], [1, 0, [a10, b10], No], [1, 2, [a10], No], [1, 4, [b10], No], [1, 6, [], Yes]];FIRST source-order witness=[[a10, b10], [c10]] Conditions: none Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample. Step identifiers and dependenciesRule: hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure · Uses: G001, S010 S012 · impliesBefore Hausdorff ORIGINAL pair index2:ORIGINAL distinct carrier points x=b10,y=c10;source open order=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []] After ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a10, b10, c10], No], [0, 2, [a10, c10], No], [0, 4, [b10, c10], No], [0, 6, [c10], No], [1, 0, [a10, b10], No], [1, 2, [a10], No], [1, 4, [b10], No], [1, 6, [], Yes]];FIRST source-order witness=[[a10, b10], [c10]] Conditions: none Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample. Step identifiers and dependenciesRule: hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure · Uses: G001, S011 S013 · impliesBefore ORIGINAL carrier order=[a10, b10, c10];ALL unordered distinct pairs with first index