**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 $X$.



Does every compact countable space contain a non-trivial convergent sequence?
Let $X=(x\_i)$ be a sequence of all elements of our space without repetitions. If it does not converge to $x\_1$, then there exists a neighbrhood $U\_1$ of $x\_1$ and a subsequence $X\_1$ of $X$ avoiding $U\_1$. If $X\_1$ does not converge to $x\_2$, then there exists a neighbrhood $U\_2$ of $x\_2$ and a subsequence $X\_2$ of $X\_1$ avoiding $U\_2$. Repeat the procedure: on the $i$th step, choose a neighborhood $U\_i$ of $x\_i$ and a subsequence $X\_i$ of $X\_{i-1}$ avoiding $U\_i$. Then the $U\_i$ cover the whole space, but any finite collection of them does not.


