Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stability of closedness of semi-algebraic sets under continuous
  semi-algebraic mappings   Given a closed semi-algebraic set $X \subset \mathbb{R}^n$ and a continuous
semi-algebraic mapping $G \colon X \to \mathbb{R}^m,$ it will be shown that
there exists an open dense semi-algebraic subset $\mathscr{U}$ of
$L(\mathbb{R}^n, \mathbb{R}^m),$ the space of all linear mappings from
$\mathbb{R}^n$ to $\mathbb{R}^m,$ such that for all $F \in \mathscr{U},$ the
image $(F + G)(X)$ is a closed (semi-algebraic) set in $\mathbb{R}^m.$ To do
this, we study the tangent cone at infinity $C_\infty X$ and the set $E_\infty
X \subset C_\infty X$ of (unit) exceptional directions at infinity of $X.$
Specifically we show that the set $E_\infty X$ is nowhere dense in $C_\infty X
\cap \mathbb{S}^{n - 1}.$

6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
14
The categories of this paper are: math-ph
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an