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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Continuous image of a compact set · qf-mod41-compactimg-reverse-implication-boundary · candidate
Canonical question JSONL · English JSONL · Validation report