# Qualified Maps, Relative Cover Pullbacks and Compact-Image Nonconverses

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-20261110-r03-continuous-image-of-compact-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.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [0, 0, 0], [0], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 0, 0], [0], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0001/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 2. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [1, 0, 0], [0, 1], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 0, 0], [0, 1], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0002/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 3. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [2, 0, 0], [0, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 0, 0], [0, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0003/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 4. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [0, 1, 0], [0, 1], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 0], [0, 1], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0004/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 5. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [1, 1, 0], [0, 1], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 1, 0], [0, 1], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0005/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 6. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [2, 1, 0], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 1, 0], [0, 1, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0006/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 7. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [0, 2, 0], [0, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 2, 0], [0, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0007/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 8. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [1, 2, 0], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 2, 0], [0, 1, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0008/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 9. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [2, 2, 0], [0, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 2, 0], [0, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0009/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 10. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [0, 0, 1], [0, 1], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 0, 1], [0, 1], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0010/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 11. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [1, 0, 1], [0, 1], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 0, 1], [0, 1], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0011/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 12. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [2, 0, 1], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 0, 1], [0, 1, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0012/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 13. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [0, 1, 1], [0, 1], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 1], [0, 1], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0013/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 14. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [1, 1, 1], [1], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 1, 1], [1], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0014/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 15. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [2, 1, 1], [1, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 1, 1], [1, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0015/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 16. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [0, 2, 1], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 2, 1], [0, 1, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0016/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 17. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [1, 2, 1], [1, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 2, 1], [1, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0017/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 18. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [2, 2, 1], [1, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 2, 1], [1, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0018/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 19. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [0, 0, 2], [0, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 0, 2], [0, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0019/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 20. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [1, 0, 2], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 0, 2], [0, 1, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0020/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 21. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [2, 0, 2], [0, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 0, 2], [0, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0021/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 22. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [0, 1, 2], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 2], [0, 1, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0022/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 23. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [1, 1, 2], [1, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 1, 2], [1, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0023/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 24. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [2, 1, 2], [1, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 1, 2], [1, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0024/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 25. The Exact Finite Image Is Separate from a Continuity Audit

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).

This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain.

View answer and explanation([0, 1, 2], [0, 2, 2], [0, 2], Yes, finite_compact_source_subset_exact_image)

Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 2, 2], [0, 2], Yes, finite_compact_source_subset_exact_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0025/)

Continuous image of a compact set · qf-mod41-compactimg-image-set · candidate





## 26. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [0, 1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], []], [[2], []]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0, 1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], []], [[2], []]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0026/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 27. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0]], [[2], []]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0]], [[2], []]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0027/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 28. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [1, 2]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [1, 2]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0028/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 29. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [0, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [1]], [[2], []]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [1]], [[2], []]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0029/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 30. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [2]], [[1], [0, 1]], [[1, 2], [0, 1]], [[2], []]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [2]], [[1], [0, 1]], [[1, 2], [0, 1]], [[2], []]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0030/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 31. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [2]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [0, 1]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [2]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [0, 1]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0031/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 32. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [0, 2]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [1]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0, 2]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [1]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0032/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 33. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [2]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0, 1]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [2]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0, 1]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0033/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 34. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [2]], [[0, 1], [2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0, 1]], [[2], [0, 1]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [2]], [[0, 1], [2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0, 1]], [[2], [0, 1]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0034/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 35. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [0, 1]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [2]], [[2], []]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0, 1]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [2]], [[2], []]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0035/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 36. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [1]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1]], [[1], [0, 2]], [[1, 2], [0, 2]], [[2], []]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [1]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1]], [[1], [0, 2]], [[1, 2], [0, 2]], [[2], []]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0036/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 37. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [1]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [0, 2]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [1]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [0, 2]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0037/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 38. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [0]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0]], [[1], [1, 2]], [[1, 2], [1, 2]], [[2], []]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0]], [[1], [1, 2]], [[1, 2], [1, 2]], [[2], []]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0038/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 39. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], []], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], []], [[1], [0, 1, 2]], [[1, 2], [0, 1, 2]], [[2], []]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], []], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], []], [[1], [0, 1, 2]], [[1, 2], [0, 1, 2]], [[2], []]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0039/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 40. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], []], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0]], [[1], [1, 2]], [[1, 2], [0, 1, 2]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], []], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0]], [[1], [1, 2]], [[1, 2], [0, 1, 2]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0040/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 41. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [0]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [1, 2]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [1, 2]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0041/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 42. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], []], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1]], [[1], [0, 2]], [[1, 2], [0, 1, 2]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], []], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1]], [[1], [0, 2]], [[1, 2], [0, 1, 2]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0042/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 43. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], []], [[0, 1], [2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [0, 1, 2]], [[2], [0, 1]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], []], [[0, 1], [2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [0, 1, 2]], [[2], [0, 1]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0043/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 44. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [0, 1]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0, 1]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0044/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 45. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [1]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0, 2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [1]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0, 2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0045/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 46. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [1]], [[0, 1], [1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0, 2]], [[2], [0, 2]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [1]], [[0, 1], [1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0, 2]], [[2], [0, 2]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0046/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 47. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [0]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [1, 2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [1, 2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0047/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 48. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], []], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [2]], [[1], [0, 1]], [[1, 2], [0, 1, 2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], []], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [2]], [[1], [0, 1]], [[1, 2], [0, 1, 2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0048/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 49. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], []], [[0, 1], [1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [0, 1, 2]], [[2], [0, 2]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], []], [[0, 1], [1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [0, 1, 2]], [[2], [0, 2]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0049/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 50. Every Target Open Must Have an Open Source Preimage

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

View answer and explanation([[[], []], [[0], [0]], [[0, 1], [0]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [1, 2]], [[2], [1, 2]]], Yes, every_target_open_has_source_open_preimage)

List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0]], [[0, 1], [0]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [1, 2]], [[2], [1, 2]]], Yes, every_target_open_has_source_open_preimage).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0050/)

Continuous image of a compact set · qf-mod41-compactimg-continuity-audit · candidate





## 51. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0], [[0]], [[0, 1, 2]], [0], [[0]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0], [[0]], [[0, 1, 2]], [0], [[0]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0051/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 52. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 1], [[0], [1]], [[1, 2], [0]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1], [[0], [1]], [[1, 2], [0]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0052/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 53. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 2], [[0], [2]], [[1, 2], [0]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 2], [[0], [2]], [[1, 2], [0]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0053/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 54. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 1], [[0], [1]], [[0, 2], [1]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1], [[0], [1]], [[0, 2], [1]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0054/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 55. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 1], [[0], [1]], [[2], [0, 1]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1], [[0], [1]], [[2], [0, 1]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0055/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 56. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 1, 2], [[0], [1], [2]], [[2], [1], [0]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[2], [1], [0]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0056/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 57. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 2], [[0], [2]], [[0, 2], [1]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 2], [[0], [2]], [[0, 2], [1]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0057/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 58. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 1, 2], [[0], [1], [2]], [[2], [0], [1]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[2], [0], [1]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0058/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 59. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 2], [[0], [2]], [[2], [0, 1]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 2], [[0], [2]], [[2], [0, 1]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0059/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 60. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 1], [[0], [1]], [[0, 1], [2]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1], [[0], [1]], [[0, 1], [2]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0060/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 61. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 1], [[0], [1]], [[1], [0, 2]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1], [[0], [1]], [[1], [0, 2]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0061/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 62. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 1, 2], [[0], [1], [2]], [[1], [2], [0]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[1], [2], [0]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0062/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 63. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 1], [[0], [1]], [[0], [1, 2]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1], [[0], [1]], [[0], [1, 2]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0063/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 64. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([1], [[1]], [[0, 1, 2]], [0], [[1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([1], [[1]], [[0, 1, 2]], [0], [[1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0064/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 65. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([1, 2], [[1], [2]], [[1, 2], [0]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([1, 2], [[1], [2]], [[1, 2], [0]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0065/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 66. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 1, 2], [[0], [1], [2]], [[0], [2], [1]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[0], [2], [1]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0066/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 67. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([1, 2], [[1], [2]], [[0, 2], [1]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([1, 2], [[1], [2]], [[0, 2], [1]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0067/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 68. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([1, 2], [[1], [2]], [[2], [0, 1]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([1, 2], [[1], [2]], [[2], [0, 1]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0068/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 69. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 2], [[0], [2]], [[0, 1], [2]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 2], [[0], [2]], [[0, 1], [2]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0069/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 70. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 1, 2], [[0], [1], [2]], [[1], [0], [2]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[1], [0], [2]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0070/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 71. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 2], [[0], [2]], [[1], [0, 2]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 2], [[0], [2]], [[1], [0, 2]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0071/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 72. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 1, 2], [[0], [1], [2]], [[0], [1], [2]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[0], [1], [2]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0072/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 73. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([1, 2], [[1], [2]], [[0, 1], [2]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([1, 2], [[1], [2]], [[0, 1], [2]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0073/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 74. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([1, 2], [[1], [2]], [[1], [0, 2]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([1, 2], [[1], [2]], [[1], [0, 2]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0074/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 75. Pull an Image Cover Back to K and Preserve Its Subcover Indices

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).

relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim.

View answer and explanation([0, 2], [[0], [2]], [[0], [1, 2]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)

Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 2], [[0], [2]], [[0], [1, 2]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0075/)

Continuous image of a compact set · qf-mod41-compactimg-cover-pullback · candidate





## 76. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-2,2) with left<right and constant map value=0 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-2, 2, 0, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-2, 2, 0, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0076/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 77. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-3,3) with left<right and constant map value=1 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-3, 3, 1, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-3, 3, 1, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0077/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 78. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-4,4) with left<right and constant map value=2 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-4, 4, 2, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-4, 4, 2, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0078/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 79. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-5,5) with left<right and constant map value=3 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-5, 5, 3, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-5, 5, 3, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0079/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 80. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

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).

View answer and explanation(-6, 6, 4, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-6, 6, 4, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0080/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 81. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-7,7) with left<right and constant map value=5 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-7, 7, 5, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-7, 7, 5, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0081/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 82. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-8,8) with left<right and constant map value=6 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-8, 8, 6, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-8, 8, 6, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0082/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 83. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-9,9) with left<right and constant map value=7 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-9, 9, 7, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-9, 9, 7, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0083/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 84. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-10,10) with left<right and constant map value=8 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-10, 10, 8, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-10, 10, 8, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0084/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 85. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-11,11) with left<right and constant map value=9 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-11, 11, 9, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-11, 11, 9, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0085/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 86. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-12,12) with left<right and constant map value=10 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-12, 12, 10, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-12, 12, 10, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0086/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 87. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-13,13) with left<right and constant map value=11 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-13, 13, 11, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-13, 13, 11, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0087/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 88. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-14,14) with left<right and constant map value=12 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-14, 14, 12, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-14, 14, 12, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0088/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 89. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-15,15) with left<right and constant map value=13 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-15, 15, 13, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-15, 15, 13, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0089/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 90. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-16,16) with left<right and constant map value=14 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-16, 16, 14, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-16, 16, 14, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0090/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 91. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-17,17) with left<right and constant map value=15 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-17, 17, 15, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-17, 17, 15, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0091/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 92. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-18,18) with left<right and constant map value=16 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-18, 18, 16, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-18, 18, 16, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0092/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 93. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-19,19) with left<right and constant map value=17 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-19, 19, 17, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-19, 19, 17, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0093/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 94. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-20,20) with left<right and constant map value=18 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-20, 20, 18, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-20, 20, 18, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0094/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 95. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-21,21) with left<right and constant map value=19 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-21, 21, 19, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-21, 21, 19, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0095/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 96. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-22,22) with left<right and constant map value=20 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-22, 22, 20, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-22, 22, 20, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0096/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 97. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-23,23) with left<right and constant map value=21 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-23, 23, 21, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-23, 23, 21, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0097/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 98. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-24,24) with left<right and constant map value=22 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-24, 24, 22, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-24, 24, 22, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0098/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 99. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-25,25) with left<right and constant map value=23 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-25, 25, 23, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-25, 25, 23, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0099/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate





## 100. A Compact Constant Image Does Not Make an Open-Interval Domain Compact

Use the ordinary real open interval (-26,26) with left<right and constant map value=24 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).

View answer and explanation(-26, 26, 24, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)

The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-26, 26, 24, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261110-r03-continuous-image-of-compact-0100/)

Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations/validation-report.json)