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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 $X$.

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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.

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  • $\begingroup$ Thank you, Ilya! In fact, this was a simplest version of the next my question. $\endgroup$
    – Taras Banakh
    Commented Nov 8, 2016 at 9:54

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