Topology-Relative Boundaries, Complements and Empty/Whole Sets
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
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-20261014-advance-20261102-r04-topological-boundary-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
2. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
3. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
4. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
5. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
6. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
7. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
8. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
9. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
10. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
11. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
12. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
13. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
14. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
15. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
16. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
17. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
18. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
19. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
20. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
21. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
22. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
23. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
24. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
25. Boundary Is Closure Minus Interior in the Same Topology
Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[0, 2]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.
View answer and explanation
([0, 2], [0, 2, 3, 4], [], [0, 2, 3, 4])
Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([0, 2], [0, 2, 3, 4], [], [0, 2, 3, 4]).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-exact · candidate
26. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2], point=0. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(0, [1, 2, 3, 4], No, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
27. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2], point=1. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(1, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
28. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3], point=3. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(3, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
29. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3], point=4. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(4, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
30. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4], point=3. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(3, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
31. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4], point=0. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(0, [1, 2, 3, 4], No, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
32. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2], point=2. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(2, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
33. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2], point=1. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(1, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
34. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3], point=1. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(1, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
35. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3], point=1. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(1, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
36. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4], point=0. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(0, [1, 2, 3, 4], No, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
37. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4], point=2. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(2, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
38. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3], point=1. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(1, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
39. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3], point=4. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(4, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
40. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3], point=4. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(4, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
41. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3], point=0. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(0, [1, 2, 3, 4], No, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
42. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4], point=3. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(3, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
43. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4], point=4. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(4, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
44. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4], point=2. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(2, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
45. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4], point=2. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(2, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
46. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4], point=0. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(0, [1, 2, 3, 4], No, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
47. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4], point=4. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(4, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
48. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4], point=3. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(3, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
49. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4], point=2. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(2, [1, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
50. Boundary Membership Is Not Just Membership in the Original Set
Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[0, 2], point=4. Give (point,boundary,in_boundary?,topology_relative_boundary).
View answer and explanation
(4, [0, 2, 3, 4], Yes, topology_relative_boundary)
Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (4, [0, 2, 3, 4], Yes, topology_relative_boundary).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-membership · candidate
51. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
52. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
53. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
54. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
55. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
56. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
57. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
58. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
59. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
60. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
61. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
62. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
63. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
64. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
65. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
66. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
67. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
68. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
69. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
70. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
71. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
72. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
73. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
74. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
75. A Set and Its Complement Have the Same Topological Boundary
Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[0, 2]. Give (subset,complement,boundary_a,boundary_complement,equal?).
View answer and explanation
([0, 2], [1, 3, 4], [0, 2, 3, 4], [0, 2, 3, 4], Yes)
Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], [0, 2, 3, 4], [0, 2, 3, 4], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-complement-invariance · candidate
76. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
77. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(whole, [0, 1, 2, 3, 4], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
78. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
79. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(whole, [0, 1, 2, 3, 4], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
80. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
81. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(whole, [0, 1, 2, 3, 4], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
82. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
83. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(whole, [0, 1, 2, 3, 4], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
84. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
85. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(whole, [0, 1, 2, 3, 4], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
86. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
87. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(whole, [0, 1, 2, 3, 4], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
88. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
89. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(whole, [0, 1, 2, 3, 4], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
90. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
91. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(whole, [0, 1, 2, 3, 4], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
92. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
93. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(whole, [0, 1, 2, 3, 4], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
94. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
95. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(whole, [0, 1, 2, 3, 4], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
96. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
97. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(whole, [0, 1, 2, 3, 4], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
98. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
99. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(whole, [0, 1, 2, 3, 4], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
100. Empty and Whole Sets Both Have Empty Boundary
Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).
View answer and explanation
(empty, [], [], Yes)
Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes).
Reviewed step-by-step derivation
Topological boundary · qf-mod33-boundary-empty-whole · candidate
Canonical question JSONL · English JSONL · Validation report