  We prove that (1) for any complete lattice $L$, the set $\mathcal{D}(L)$ of
all nonempty saturated compact subsets of the Scott space of $L$ is a complete
Heyting algebra (with the reverse inclusion order); and (2) if the Scott space
of a complete lattice $L$ is non-sober, then the Scott space of
$\mathcal{D}(L)$ is non-sober. Using these results and the Isbell's example of
a non-sober complete lattice, we deduce that there is a complete Heyting
algebra whose Scott space is non-sober, thus give a positive answer to a
problem posed by Jung. We will also prove that a $T_0$ space is well-filtered
iff its upper space (the set $\mathcal{D}(X)$ of all nonempty saturated compact
subsets of $X$ equipped with the upper Vietoris topology) is well-filtered,
which answers another open problem.
