Complete Finite Topology Axioms and Open-Set Inventories
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261018-r01-topological-space-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
2. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [0, 1], [0, 1, 2]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [1], [0, 1], [0, 1, 2]], Yes, No, Yes, Yes, No, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [0, 1], [0, 1, 2]], Yes, No, Yes, Yes, No, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
3. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [0, 2], [0, 1, 2], [3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [2], [0, 2], [0, 1, 2], [3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[2], [3], [2, 3]], none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [0, 2], [0, 1, 2], [3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[2], [3], [2, 3]], none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
4. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, Yes, Yes, Yes, Yes, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, Yes, Yes, Yes, Yes, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
5. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[0], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
6. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [1, 2], [0, 1, 2]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [1], [1, 2], [0, 1, 2]], Yes, No, Yes, Yes, No, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 2], [0, 1, 2]], Yes, No, Yes, Yes, No, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
7. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [1, 2], [0, 1, 2], [3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [2], [1, 2], [0, 1, 2], [3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[2], [3], [2, 3]], none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [1, 2], [0, 1, 2], [3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[2], [3], [2, 3]], none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
8. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, Yes, Yes, Yes, Yes, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, Yes, Yes, Yes, Yes, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
9. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[0], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
10. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [1, 3], [0, 1, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [1], [1, 3], [0, 1, 3]], Yes, No, Yes, Yes, No, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 3], [0, 1, 3]], Yes, No, Yes, Yes, No, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
11. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [1], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[1], [2], [1, 2]], none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[1], [2], [1, 2]], none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
12. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, Yes, Yes, Yes, Yes, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, Yes, Yes, Yes, Yes, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
13. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
14. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [0, 1], [0, 1, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [1], [0, 1], [0, 1, 3]], Yes, No, Yes, Yes, No, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [0, 1], [0, 1, 3]], Yes, No, Yes, Yes, No, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
15. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [1], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[1], [2], [1, 2]], none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[1], [2], [1, 2]], none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
16. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, Yes, Yes, Yes, Yes, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, Yes, Yes, Yes, Yes, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
17. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[0], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
18. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [1, 2], [1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [1], [1, 2], [1, 2, 3]], Yes, No, Yes, Yes, No, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 2], [1, 2, 3]], Yes, No, Yes, Yes, No, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
19. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [0], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[0], [2], [0, 2]], none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[0], [2], [0, 2]], none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
20. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, Yes, Yes, Yes, Yes, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, Yes, Yes, Yes, Yes, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
21. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
22. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [1, 3], [1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [1], [1, 3], [1, 2, 3]], Yes, No, Yes, Yes, No, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 3], [1, 2, 3]], Yes, No, Yes, Yes, No, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
23. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [0], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[0], [2], [0, 2]], none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[0], [2], [0, 2]], none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
24. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(4, [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, Yes, Yes, Yes, Yes, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, Yes, Yes, Yes, Yes, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
25. Complete Finite Topology Axiom Audit
Complete finite data: X=Fin(3)={0,…,2}. Candidate open family=[[], [0], [0, 1], [0, 1, 2]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).
View answer and explanation
(3, [[], [0], [0, 1], [0, 1, 2]], Yes, Yes, Yes, Yes, Yes, none, none)
Check every ordered pair, including diagonal pairs, and the empty/full requirements separately. Finiteness makes the complete family check sufficient for this finite model, not a general formal proof. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[], [0], [0, 1], [0, 1, 2]], Yes, Yes, Yes, Yes, Yes, none, none).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-axiom-audit · candidate
26. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
27. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
28. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
29. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
30. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
31. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
32. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
33. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
34. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
35. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
36. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
37. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
38. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
39. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
40. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
41. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
42. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
43. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
44. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
45. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
46. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
47. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
48. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
49. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(4, [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
50. Every Open, Closed, and Clopen Set in a Finite Topology
Complete finite data: X=Fin(3)={0,…,2}. Candidate open family=[[], [0], [0, 1], [0, 1, 2]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.
View answer and explanation
(3, [[], [0], [0, 1], [0, 1, 2]], [[], [2], [1, 2], [0, 1, 2]], [[], [0, 1, 2]])
Take all complements in the fixed underlying set and intersect the resulting closed family with the open family. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[], [0], [0, 1], [0, 1, 2]], [[], [2], [1, 2], [0, 1, 2]], [[], [0, 1, 2]]).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-open-closed-inventory · candidate
51. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, discrete)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, discrete).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
52. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [0, 1, 2, 3]], Yes, indiscrete)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], Yes, indiscrete).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
53. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
54. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
55. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
56. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
57. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
58. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
59. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
60. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
61. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
62. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
63. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
64. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
65. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
66. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
67. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
68. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
69. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
70. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
71. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
72. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
73. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
74. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
75. Topology Qualification before Discrete or Indiscrete Classification
Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.
View answer and explanation
(4, [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, other-topology)
Qualify the candidate before classification; the discrete family is the entire powerset, whereas the indiscrete family contains only the empty set and X. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, other-topology).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-discrete-indiscrete · candidate
76. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
77. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
78. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
79. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
80. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
81. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
82. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
83. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
84. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
85. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
86. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
87. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
88. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
89. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
90. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
91. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
92. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
93. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
94. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
95. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
96. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
97. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
98. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
99. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0, 1, 2, 3]], [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0, 1, 2, 3]], [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
100. Compare Two Qualified Topologies on the Same Finite Set
Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], B=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.
View answer and explanation
(4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser)
Both candidates must satisfy the topology axioms. A⊆B means B is finer; equality and incomparability are distinct outcomes. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser).
Reviewed step-by-step derivation
Topological space · qf-ext18-topology-finer-coarser · candidate
Canonical question JSONL · English JSONL · Validation report