Connected Topological Spaces: Actual Spaces and Independent Reference Conventions
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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-20261103-r05-connected-topological-space-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.
1. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 1], [2, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
2. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 1], [2, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
3. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 1], [2, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
4. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 1], [2, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
5. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 1], [2, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
6. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 1], [2, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
7. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 2], right=[1, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 2], [1, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
8. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 2], right=[1, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 2], [1, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
9. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 2], right=[1, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 2], [1, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
10. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 2], right=[1, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 2], [1, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
11. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 2], right=[1, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 2], [1, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
12. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 2], right=[1, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 2], [1, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
13. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 3], right=[1, 2, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 3], [1, 2, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
14. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 3], right=[1, 2, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 3], [1, 2, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
15. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 3], right=[1, 2, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 3], [1, 2, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
16. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 3], right=[1, 2, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 3], [1, 2, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
17. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 3], right=[1, 2, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 3], [1, 2, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
18. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 3], right=[1, 2, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 3], [1, 2, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
19. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 4], right=[1, 2, 3]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 4], [1, 2, 3], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
20. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 4], right=[1, 2, 3]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 4], [1, 2, 3], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
21. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 4], right=[1, 2, 3]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 4], [1, 2, 3], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
22. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 4], right=[1, 2, 3]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 4], [1, 2, 3], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
23. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 4], right=[1, 2, 3]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 4], [1, 2, 3], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
24. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 4], right=[1, 2, 3]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 4], [1, 2, 3], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
25. A Separation Must Meet Every Condition on the Actual Space
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Proposed left=[0, 1], right=[2, 3, 4]. Give (left,right,valid_separation?,classification), with classification=disconnected_separation or not_a_separation.
View answer and explanation
([0, 1], [2, 3, 4], No, not_a_separation)
Check openness,nonemptiness,disjointness and full coverage together. A valid proposal witnesses disconnection. An invalid proposal is merely not a separation; it does not establish connectedness of the whole space. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, not_a_separation).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-separation-audit · candidate
26. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
27. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
28. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
29. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
30. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
31. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
32. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 2]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
33. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 2]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
34. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 2]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
35. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 2]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
36. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 2]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
37. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 2]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
38. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 3]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
39. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 3]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
40. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 3]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
41. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 3]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
42. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 3]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
43. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 3]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
44. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 4]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
45. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 4]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
46. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 4]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
47. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 4]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
48. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 4]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
49. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 4]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
50. A Clopen Witness Must Also Be Nontrivial
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Selected subset=[0, 1]. Give (subset,complement,clopen?,nontrivial?,disconnects?,nontrivial_clopen_witness).
View answer and explanation
([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness)
Use the complement inside the actual carrier. Both sets must be open,and the selected subset must be neither empty nor whole. The disconnects flag conjoins these conditions; a failed selected witness is not an exhaustive connectedness proof. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], No, Yes, No, nontrivial_clopen_witness).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-clopen-witness · candidate
51. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
52. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
53. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
54. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
55. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
56. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
57. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
58. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
59. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
60. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
61. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
62. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
63. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
64. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
65. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
66. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
67. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
68. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
69. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
70. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
71. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
72. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
73. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
74. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
75. Exhaust the Complete Finite Topology before Certifying Connectedness
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Give (clopen_sets,nontrivial_clopen,connected?,exhaustive_complete_finite_topology), retaining the supplied topology order within both lists.
View answer and explanation
([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology)
Check the complement of every declared open subset and list every clopen set. Remove empty and whole sets to form the nontrivial list. Exactly an empty nontrivial list certifies connectedness under this explicit convention. The exact result, retaining the requested order, is ([[], [0, 1, 2, 3, 4]], [], Yes, exhaustive_complete_finite_topology).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-exhaustive · candidate
76. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
77. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(singleton, 5, Yes, singleton_connected)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
78. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
79. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(singleton, 5, Yes, singleton_connected)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
80. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
81. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(singleton, 5, Yes, singleton_connected)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
82. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
83. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(singleton, 5, Yes, singleton_connected)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
84. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
85. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(singleton, 5, Yes, singleton_connected)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
86. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
87. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(singleton, 5, Yes, singleton_connected)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
88. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
89. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(singleton, 5, Yes, singleton_connected)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
90. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
91. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(singleton, 5, Yes, singleton_connected)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
92. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
93. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(singleton, 5, Yes, singleton_connected)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
94. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
95. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(singleton, 5, Yes, singleton_connected)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
96. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
97. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(singleton, 5, Yes, singleton_connected)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
98. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
99. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=singleton. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(singleton, 5, Yes, singleton_connected)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (singleton, 5, Yes, singleton_connected).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
100. Separate the Actual Finite Space from an Independent Reference Convention
Use the actual finite carrier Fin(5)={0,…,4} and its complete declared topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], with subsets in the supplied order. Empty and whole sets, all finite intersections and all unions must belong to this finite topology. Connectedness means absence of a separation into two disjoint nonempty open sets covering this actual carrier, equivalently absence of a nontrivial clopen subset. It is not graph connectivity or a conclusion from selected tests. The empty-space convention here is explicitly the vacuous no-separation convention,not a nonempty ConnectedSpace type. Independent reference_kind=empty. This label selects only the empty-space or singleton reference convention; it does not say that the actual 5-point carrier equals that reference. Give (reference_kind,actual_carrier_size,actual_connected?,reference_convention). reference_convention is empty_space_connected_by_declared_vacuous_convention or singleton_connected according to the independent reference label.
View answer and explanation
(empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention)
Compute actual_carrier_size and actual_connected only from the actual complete finite topology. Report the separate named reference convention without using it as a certificate for this actual space. Even a disconnected actual space can be compared with the true empty/singleton reference convention. The exact result, retaining the requested order, is (empty, 5, Yes, empty_space_connected_by_declared_vacuous_convention).
Reviewed step-by-step derivation
Connected topological space · qf-mod34-connected-empty-singleton-boundary · candidate
Canonical question JSONL · English JSONL · Validation report