Uncountable Finite Complement Space is not First-Countable

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

Then $T$ is not first-countable

Proof
Suppose some $x \in S$ has a countable local basis.

That means:
 * there exists a set $\mathcal B_x \subseteq \tau$

such that:
 * $\forall B \in \mathcal B_x: x \in B$

and such that:
 * every open neighborhood of $x$ contains some $B \in \mathcal B_x$.

So:

By definition, each of $S \setminus B$ is finite.

From Countable Union of Countable Sets is Countable it follows that $\displaystyle \bigcup_{B \mathop \in \mathcal B_x} \left({S \setminus B}\right)$ is countable.

So $S \setminus \left\{{x}\right\}$ and therefore $S$ is also countable.

From this contradiction (as we have specified that $S$ is uncountable) it follows that our assumption that $x \in S$ has a countable local basis must be false.

Hence by definition $T$ can not be first-countable.