Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The upper Vietoris topology on the space of inverse-closed subsets of a
  spectral space and applications   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 $D$.

15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom