Let $X$ be as non-Hausdorff compactification of $\mathbb{Z}$ with two points $\infty_1$ and $\infty_2$ such that a basis of neighborhoods of$\infty_i$ is formed by the sets $U$, $\infty_i\in U$, with finite complement.
$X$ is a compact non-Hausdorff space. What can say about the first countability of it.
Is it true that $X$ is first countable?
Please help me to know it.
$X$ is countable, so it has countably many finite subsets and so the given local base of both $\infty_i$ is countable already. Every point $n$ of $\Bbb Z$ has $\{\{n\}\}$ as a local base (being a discrete space). Draw your conclusions.