Original question and givens
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.
G001 — 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.
Step-by-step derivation
S001 · implies
Before
ACTUAL carrier Fin(5);original ordered complete topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]
After
unique empty/whole opens present;empty carrier allowed;first union failure=null;first intersection failure=null
Conditions: none
The actual n may be0,when empty and whole are the same open rather than two distinct sets.Check canonical subsets and all binary union/intersection obligations;the finite family then handles arbitrary unions.This data fixes the actual space,not a reference label.
Step identifiers and dependencies
Rule: connected_actual_complete_topology_including_empty_carrier · Uses: G001
S002 · implies
Before
original open index=0;open=[];ACTUAL same-carrier complement=[0, 1, 2, 3, 4]
After
complement open=Yes;nonempty proper subset=No;nontrivial clopen obstruction=No
Conditions: none
The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness.
Step identifiers and dependencies
Rule: connected_every_actual_open_complement_not_selected_samples · Uses: G001, S001
S003 · implies
Before
original open index=1;open=[0];ACTUAL same-carrier complement=[1, 2, 3, 4]
After
complement open=No;nonempty proper subset=Yes;nontrivial clopen obstruction=No
Conditions: none
The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness.
Step identifiers and dependencies
Rule: connected_every_actual_open_complement_not_selected_samples · Uses: G001, S002
S004 · implies
Before
original open index=2;open=[0, 4];ACTUAL same-carrier complement=[1, 2, 3]
After
complement open=No;nonempty proper subset=Yes;nontrivial clopen obstruction=No
Conditions: none
The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness.
Step identifiers and dependencies
Rule: connected_every_actual_open_complement_not_selected_samples · Uses: G001, S003
S005 · implies
Before
original open index=3;open=[0, 2, 4];ACTUAL same-carrier complement=[1, 3]
After
complement open=No;nonempty proper subset=Yes;nontrivial clopen obstruction=No
Conditions: none
The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness.
Step identifiers and dependencies
Rule: connected_every_actual_open_complement_not_selected_samples · Uses: G001, S004
S006 · implies
Before
original open index=4;open=[0, 1, 2, 4];ACTUAL same-carrier complement=[3]
After
complement open=No;nonempty proper subset=Yes;nontrivial clopen obstruction=No
Conditions: none
The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness.
Step identifiers and dependencies
Rule: connected_every_actual_open_complement_not_selected_samples · Uses: G001, S005
S007 · implies
Before
original open index=5;open=[0, 1, 2, 3, 4];ACTUAL same-carrier complement=[]
After
complement open=Yes;nonempty proper subset=No;nontrivial clopen obstruction=No
Conditions: none
The original member is open;it is clopen only if its complement is open.A separation obstruction also requires nonemptiness and properness.List all candidates in original topology order;one failed candidate cannot establish connectedness.
Step identifiers and dependencies
Rule: connected_every_actual_open_complement_not_selected_samples · Uses: G001, S006
S008 · implies
Before
ALL actual clopen sets=[[], [0, 1, 2, 3, 4]];ALL nontrivial clopen sets=[]
After
a nontrivial clopen S gives separation(S,X minus S);a separation by disjoint nonempty opens gives such a clopen S;actual connected iff no such set
Conditions: none
A proper nonempty clopen and its open complement separate the carrier.Conversely the two opens in a separation are complements,so either is nontrivial clopen.Exhausting the complete finite topology therefore excludes every separation.This is not connectivity of an unrelated graph.
Step identifiers and dependencies
Rule: connected_clopen_separation_equivalence_full_finite_exhaustion · Uses: G001, S007
S009 · implies
Before
selected ACTUAL left=[0, 4];right=[1, 2, 3];open flags=[Yes, No];nonempty flags=[Yes, Yes];disjoint=Yes;covers whole actual carrier=Yes
After
all separation conditions=No;classification=not_a_separation
Conditions: none
Both sides must be open,nonempty,disjoint and cover the actual carrier.A valid proposal witnesses disconnection;failure of this proposal alone is not a connectedness proof.
Step identifiers and dependencies
Rule: connected_selected_separation_all_conditions_not_global_converse · Uses: G001, S008