= Countable unions of Ellentuck clopen sets need not be closed
In the <space of infinite subsets of the natural numbers>, the families $C_n=\{X:n\notin X\}$ are <clopen sets> in the <ordinary topology on infinite subsets> and thus in the finer <Ellentuck topology>. Their union is $[\mathbb N]^\omega\setminus\{\mathbb N\}$. Every basic <Ellentuck topology> neighborhood of $\mathbb N$ contains $\mathbb N\setminus\{n\}$ for some sufficiently large $n$, so this union is not a <closed set>.
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