Is a connected, first countable space necessarily Hausdorff?

Question

I've been trying for forever to come up with a counterexample but haven't had any luck.

Answer 1

The Sierpinski Two-Point Space is connected and first countable, but not Hausdorff (indeed, not even $T_1$).

Answer 2

Another example: the cofinite topology on the integers is $T_1$, second countable (so first countable), connected and very non-Hausdorff for the same reason: all non-empty open sets intersect.