**Problem.** Does every compact countable space contain a non-trivial convergent sequence?

This question concerns non-Hausdorff compact spaces. An example of such space is any infinite set $X$ endowed with the Zariski topology $\tau=\{\emptyset\}\cup\{X\setminus F:F$ is finite$\}$. Observe that this space is compact, topologically homogeneous, contains no isolated points, but each sequence $(x_n)_{n\in\omega}$ of pairwise distinct points of $X$ converges to any point of $X$.