Non-Hausdorff Space With Only One Limit

Question

I'm looking for an example of a non-Hausdorff topological space where every convergent sequence converges to the same point. In other words, if $a_n$ and $b_n$ are convergent sequences in $X$, then they both converge to the point $x \in X$.

Does one exist?

Answer 1

Let $X=\{0,1\}$ with the indiscrete topology. Then any sequence $a_n$ converges to $0$.