  Given an arbitrary spectral space $X$, we consider the set
${\boldsymbol{\mathcal{X}}}(X)$ of all nonempty subsets of $X$ that are closed
with respect to the inverse topology. We introduce a Zariski-like topology on
${\boldsymbol{\mathcal{X}}}(X)$ and, after observing that it coincides the
upper Vietoris topology, we prove that ${\boldsymbol{\mathcal{X}}}(X)$ is
itself a spectral space, that this construction is functorial, and that
${\boldsymbol{\mathcal{X}}}(X)$ provides an extension of $X$ in a more
`complete' spectral space. Among the applications, we show that, starting from
an integral domain $D$, ${\boldsymbol{\mathcal{X}}}(\mathrm{Spec}(D))$ is
homeomorphic to the (spectral) space of all the stable semistar operations of
finite type on $D$.
