= Solution
Take the <cone on an infinite coinfinite ground set>
$$
\boxed{E=[2\mathbb N]^\omega.}
$$
It is $*$-open because $E=[\varnothing,2\mathbb N]$ is a basic <open set> of the <Ellentuck topology>. It is not $\tau$-open: if $X\in E$, every ordinary basic <neighbourhood> $[s]$ of $X$ contains an infinite extension of $s$ having an odd point larger than $\max s$. That extension is outside $E$. Therefore no point of $E$ has an ordinary open <neighbourhood> contained in $E$.
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