Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 4, 1, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).
G001 — Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 4, 1, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).
Definition and scope
Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope.
original complete carrier Fin(5);original TOTAL source-indexed candidate map=[0, 3, 4, 1, 2]
After
all target values distinct and complete;explicit inverse=[0, 3, 4, 1, 2];inverse(map(x))=x and map(inverse(y))=y for EVERY carrier point
Conditions: none
Accept a bijection only when every target appears exactly once in the complete source list.Build the inverse by original indices and retain both inverse identities rather than trusting a family name.
ACTUAL finite carriers Fin(5);complete bijection=[0, 3, 4, 1, 2] and inverse=[0, 3, 4, 1, 2];BOTH topologies declared discrete
After
for ANY subset on either target,its preimage is a subset and therefore open;both map and inverse continuous;homeomorphism in this finite discrete scope
Conditions: none
Every subset is open in a discrete topology,so preimages of arbitrary subsets are open in both directions.This uses both declared discrete topologies,not an arbitrary-topology bijection,and supplies no metric or differentiable conclusion.
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.