# Question 22080: A Compact Constant Image Does Not Make an Open-Interval Domain Compact

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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.

R0043-22080 · q-20261014-advance-20261110-r03-continuous-image-of-compact-0080



## Original question and givens

Use the ordinary real open interval (-6,6) with left<right and constant map value=4 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).


- G001 — Use the ordinary real open interval (-6,6) with left<right and constant map value=4 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, 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-continuous-image-of-compact](https://ocme.evemisslab.com/en/mko/mko-continuous-image-of-compact/)





## Step-by-step derivation

S001 · impliesBefore

source operation=compactimg-reverse-implication-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary]

Conditions: none

Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL WHOLE real interval(-6,6);constant real image value=4

After

for integers j>=0,relative opens U_j=(L+(R-L)/(j+2),R) cover the interval;ANY finite subfamily is contained in U_max_j and misses a nonempty interval near L,so domain NOT compact

Conditions: none

For every x>L,the Archimedean property gives j whose threshold is below x,so the whole family covers.Any finite family has a largest j and misses points between L and its positive threshold.This is an actual cover argument,not finite sampling or a nonclosed slogan.

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





S003 · impliesBefore

ORIGINAL noncompact domain=(-6,6);constant f(x)=4

After

preimage of ANY target open is either the whole domain or empty,so continuous;image singleton has a one-member subcover from any cover;reverse implication claim=No

Conditions: none

Both possible preimages of a constant map are open.A cover of a singleton has a member containing its point.Thus a compact image does not force a compact domain,nor general compact preimages or closed images without further hypotheses.

Step identifiers and dependenciesRule: compactimg_constant_map_continuity_singleton_compact_no_converse · 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

(-6, 6, 4, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

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: 5ef786de8beca6f73749471ab4e25e2bd0d330be67fa363a4fc9ed2d5924448a
- candidate record SHA-256: ae5121b73eaa31e8eb357e7c8db87b1243fc50045b8f9268e64c9cc43726ae69
- semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
- question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0080.json)