Countable Finite Complement Space is not Locally Path-Connected

Theorem
Let $T = \left({S, \tau}\right)$ be a finite complement topology on a countable set $S$.

Then $T$ is not locally path-connected.

Proof
Let $\mathcal B$ be a basis for $T$.

Let $B \in \mathcal B$.

Let $f: \left[{0 .. 1}\right] \to B$ be a path on $T$.

Then $f$ is by definition continuous.

But consider the set:
 * $F = \left\{{f^{-1} \left({x}\right): x \in B}\right\}$

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.

But $\displaystyle \bigcup F = \left[{0 .. 1}\right]$.

So $F$ is a countable set of pairwise disjoint closed sets whose union is $\left[{0 .. 1}\right]$.

This is impossible.

From this contradiction, $f$ can not be continuous, and so cannot be a path on $B$.

So $B$ cannot be path-connected.

So, by definition, $T$ is not locally path-connected.