derivation-mko-topological-connected-component-01 · ocme-new142-reader-adoption-20261021
Find every component of X={a,b,c}, tau={empty,{a},{b,c},X}.
Use each induced topology, not a graph; do not extrapolate finite clopen components.
setstopologydefinition_hereG1 — X={a,b,c}, tau={empty,{a},{b,c},X}; the empty space is connected by convention.C_x — component containing x\mathcal C — all connected subsetsmko_definition · cited — Finite families enumerate every induced subspace and connected subset. The union of all connected subsets containing x is its unique inclusion-maximal component. Then check that the components are pairwise equal or disjoint, cover X and are closed in the stated finite model. Finitely many closed components also make each component open, a finite-specific conclusion. In a general space components need not be open: in the rational subspace of the real line they are singletons and are not open. Do not substitute globally largest cardinality, graph components or a different structure on the same carrier. · mko-topological-connected-componentsets · prerequisite — Compute unions, intersections, complements, inclusion and preimages element by element.topology · prerequisite — A topology contains empty and whole sets and is closed under arbitrary unions and finite intersections; a finite family admits exhaustive checks.definition_here · prerequisite — Subspace opens are ambient-open intersections; a connected subset has no nontrivial relative-open separation; the component of x is the union of all connected subsets containing x, equivalently the inclusion-maximal connected subset.Before
\{a\}\cap\{b,c\}=\varnothing,\quad\{a\}\cup\{b,c\}=XAfter
\tau\text{ is a topology}Conditions: none
Rule: topology · Uses: G1
Leaf basis: topology
Apart from the complementary pair, operations with empty and whole stay in the family; every union/intersection axiom follows.
Before
\mathcal P(X)
After
\varnothing,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},XConditions: none
Rule: definition_here · Uses: G1
Leaf basis: definition_here
List all eight subsets before computing their induced topologies.
Before
Y=\varnothing,\{a\},\{b\},\{c\}After
\tau_\varnothing=\{\varnothing\},\quad\tau_{\{p\}}=\{\varnothing,\{p\}\}Conditions: none
Rule: sets · Uses: S01, S02
Leaf basis: sets
The four small subsets have the listed induced topologies and no nontrivial separation.
Before
Y=\{a,b\},\{a,c\}After
\tau_{\{a,b\}}=\mathcal P(\{a,b\}),\quad\tau_{\{a,c\}}=\mathcal P(\{a,c\})Conditions: none
Rule: sets · Uses: S01, S02
Leaf basis: sets
Ambient intersections give two-point discrete topologies, each separated by its two singletons and therefore disconnected.
Before
Y=\{b,c\},\ XAfter
\tau_{\{b,c\}}=\{\varnothing,\{b,c\}\};\quad X=\{a\}\sqcup\{b,c\}Conditions: none
Rule: sets · Uses: S01, S02
Leaf basis: sets
The induced topology on {b,c} is indiscrete and connected; the two complementary nonempty opens explicitly separate X.
Before
\mathcal C=\{\varnothing,\{a\},\{b\},\{c\},\{b,c\}\}After
C_a=\bigcup\{A\in\mathcal C:a\in A\}=\{a\};\quad C_b=C_c=\{b,c\}Conditions: none
Rule: definition_here · Uses: S03, S04, S05
Leaf basis: definition_here
Unite all connected subsets containing the given point, obtaining the unique inclusion-maximal candidates.
Before
\{C_a,C_b\}=\{\{a\},\{b,c\}\}After
C_a\cap C_b=\varnothing,\quad C_a\cup C_b=X
Conditions: none
Rule: sets · Uses: S06
Leaf basis: sets
The distinct components are disjoint and cover the entire carrier.
Before
X\setminus C_a=C_b\in\tau,\quad X\setminus C_b=C_a\in\tau
After
C_a,C_b\text{ both closed and open}Conditions: none
Rule: sets · Uses: G1, S07
Leaf basis: sets
Both components are closed and open in this finite model; this does not assert general components are open.
No substeps in this guide.
S08 · premise_audit · G1, S07 — In Q components are singletons but not open: an intermediate irrational separates any two rationals, while every rational neighborhood contains other rationals.S07 · premise_audit · S06 — The duplicate C_c=C_b is normalized, not counted as a third component.S06 · premise_audit · S03, S04, S05 — The smaller {a} is maximal: the larger {b,c} does not contain it.S05 · premise_audit · S01, S02 — Connectedness has now been decided for all eight subsets.S04 · premise_audit · S01, S02 — The complete four-set power sets are specified, rather than inferred from cardinality.S03 · premise_audit · S01, S02 — The declared convention handles empty, and a singleton has only one nonempty set.S02 · premise_audit · G1 — Maximality must use all candidates, not just large-looking sets.S01 · premise_audit · G1 — The four-member list is complete.Check all eight induced subspaces before inclusion-maximality.
Maximal means no proper connected superset, not largest cardinality.
reject — {a} is smaller but remains inclusion-maximal.reject — Its induced two-point discrete topology has a separation.reject — The singleton components of Q are not open, as justified at S08.The components {a} and {b,c} partition X and are clopen in this finite space.
0 declared checks for this guide; only the listed expressions are covered.
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