= Solution
For $n\geq1$, let $C_n=\{Z\in X:n\notin Z\}$. The presence or absence of $n$ is determined by a sufficiently long finite <initial segment>, so $C_n$ and its complement are <open sets> in the <ordinary topology on infinite subsets>. The <Ellentuck topology> is finer, hence each $C_n$ is star-<clopen> and in particular star-<closed>. Their union is
$$
\bigcup_{n\geq1}C_n=X\setminus\{\mathbb N\}.
$$
Every basic star-neighborhood $[s,A]$ of $\mathbb N$ has $s=\{1,\ldots,k\}$ for some $k\geq0$, and its reservoir $A$ contains every integer greater than $k$. Choosing $n>k$ gives $\mathbb N\setminus\{n\}\in[s,A]\cap C_n$. Thus $\mathbb N$ is in the star-<closure> of the union, but not in the union itself. Therefore
$$
\boxed{\bigcup_{n\geq1}C_n\text{ is not star-closed, although every }C_n\text{ is star-closed}.}
$$
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