= Solution
The answer is \b[no]. Let $X_0=\mathbb N$, regarded as one point of the <space of infinite subsets of the natural numbers>, and put
$$
O=[\mathbb N]^\omega\setminus\{X_0\}
=\bigcup_{j\geq1}[\varnothing,\mathbb N\setminus\{j\}].
$$
The displayed union makes $O$ an <open set> in the <Ellentuck topology>. On the other hand, every basic <neighbourhood> $[s,A]$ of $X_0$ contains a different infinite set: keep $s$ and remove one element from its infinite remaining tail. This other point belongs to $O$. Therefore $X_0$ is in the <closure> of $O$ but not in $O$, so
$$
\boxed{O\text{ is star-open but not star-closed}.}
$$
The fact that basic <neighbourhoods> are <clopen sets> does not make arbitrary unions of them closed.
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