# Question 19050: A Finite Discrete Bijection Is a Homeomorphism within That Scope

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0037-19050 · q-20261014-advance-20261105-r05-homeomorphism-0075



## Original question and givens

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[1, 0, 2, 3, 4]. 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=[1, 0, 2, 3, 4]. 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.


- [mko-homeomorphism](https://ocme.evemisslab.com/en/mko/mko-homeomorphism/)





## Step-by-step derivation

S001 · impliesBefore

source operation=homeo-finite-discrete-bijection;original snapshot and mathematics retained

After

explicit ordered source output fields=[vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary]

Conditions: none

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.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

original complete carrier Fin(5);original TOTAL source-indexed candidate map=[1, 0, 2, 3, 4]

After

all target values distinct and complete;explicit inverse=[1, 0, 2, 3, 4];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.

Step identifiers and dependenciesRule: homeo_complete_bijection_not_name_assumption · Uses: G001, S001





S003 · impliesBefore

ACTUAL finite carriers Fin(5);complete bijection=[1, 0, 2, 3, 4] and inverse=[1, 0, 2, 3, 4];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.

Step identifiers and dependenciesRule: homeo_discrete_all_subsets_two_way_not_general_bijection · Uses: G001, S002









## Reverse checks


- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(5, [1, 0, 2, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: fabbf64cf9a301d1413ca94e083e13030dc4d5b084f950239ea5822cc5fc6fac
- candidate record SHA-256: c970f880d9fcaef579daae897d0009091ce2a0328e7037e18cc486d04081e619
- semantic review receipt SHA-256: 065507b602ceee60cf47a700d387a51940e0c6724c35a3336a7a7948a199ed80
- question-first witness SHA-256: 7203f33b030792559673b0fee520a8c0e70903c4dd970b35293aa40852f333d6
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r05-homeomorphism-0075.json)