Countable Finite Complement Space is not Path-Connected
Theorem
Let $T = \struct {S, \tau}$ be a countable finite complement topology.
Then $T$ is not path-connected.
Proof
Aiming for a contradiction, suppose $f: \closedint 0 1 \to S$ is a path on $T$.
Then $f$ is by definition continuous.
By definition of the finite complement topology, for all $x \in S$, the set $\set x$ is closed.
Now consider the set:
- $F = \set {\map {f^{-1} } x: x \in S}$
From Continuity Defined from Closed Sets, each of the elements of $F$ is closed.
Also, from Mapping Induces Partition on Domain, the elements of $F$ are pairwise disjoint.
As $T$ is a countable finite complement topology, its underlying set $S$ is countably infinite by definition.
Hence $F$ is also countably infinite by nature of $f$ being a mapping.
Furthermore, we have $\ds \bigcup F = \closedint 0 1$.
Hence $F$ is a countably infinite set of pairwise disjoint closed sets whose union is $\closedint 0 1$.
From Closed Unit Interval is not Countably Infinite Union of Disjoint Closed Sets, this is impossible.
From this contradiction, $f$ cannot be continuous, and so cannot be a path on $T$.
So $T$ cannot be path-connected.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $18$. Finite Complement Topology on a Countable Space: $10$