Original question and givens
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.
G001 — 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.
Step-by-step derivation
S001 · implies
Before
ACTUAL carrier Fin(5);original ordered complete topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 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, 3, 4];ACTUAL same-carrier complement=[1, 2]
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, 2, 3, 4];ACTUAL same-carrier complement=[1]
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
INDEPENDENT reference_kind=singleton;ACTUAL carrier size=5;ACTUAL complete-topology connected=Yes
After
reference_convention=singleton_connected;reference label does NOT assert actual carrier is the reference space
Conditions: none
The label selects an independent empty/singleton reference convention.It neither forces actual n to0/1 nor certifies the actual space using the reference fact.Actual connectivity comes only from the complete actual topology.Empty-space connectedness uses the vacuous no-separation convention,not a nonempty Lean ConnectedSpace claim.
Step identifiers and dependencies
Rule: connected_reference_convention_never_rewrites_actual_space · Uses: G001, S008