Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on hyperspaces by closed sets with Vietoris topology   For a topological space $X$, let $CL(X)$ be the set of all non-empty closed
subset of $X$, and denote the set $CL(X)$ with the Vietoris topology by
$(CL(X), \mathbb{V})$. In this paper, we mainly discuss the hyperspace $(CL(X),
\mathbb{V})$ when $X$ is an infinite countable discrete space. As an
application, we first prove that the hyperspace with the Vietoris topology on
an infinite countable discrete space contains a closed copy of $n$-th power of
Sorgenfrey line for each $n\in\mathbb{N}$. Then we investigate the tightness of
the hyperspace $(CL(X), \mathbb{V})$, and prove that the tightness of $(CL(X),
\mathbb{V})$ is equal to the set-tightness of $X$. Moreover, we extend some
results about the generalized metric properties on the hyperspace $(CL(X),
\mathbb{V})$. Finally, we give a characterization of $X$ such that $(CL(X),
\mathbb{V})$ is a $\gamma$-space.

25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
24
The categories of this paper are: dg-ga
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
17
The categories of this paper are: q-fin
28
The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg