Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ascoli and sequentially Ascoli spaces   A Tychonoff space $X$ is called ({\em sequentially}) {\em Ascoli} if every
compact subset (resp. convergent sequence) of $C_k(X)$ is evenly continuous,
where $C_k(X)$ denotes the space of all real-valued continuous functions on $X$
endowed with the compact-open topology. Various properties of (sequentially)
Ascoli spaces are studied, and we give several characterizations of
sequentially Ascoli spaces. Strengthening a result of Arhangel'skii we show
that a hereditary Ascoli space is Fr\'{e}chet--Urysohn. A locally compact
abelian group $G$ with the Bohr topology is sequentially Ascoli iff $G$ is
compact. If $X$ is totally countably compact or near sequentially compact then
it is a sequentially Ascoli space. The product of a locally compact space and
an Ascoli space is Ascoli. If additionally $X$ is a $\mu$-space, then $X$ is
locally compact iff the product of $X$ with any Ascoli space is an Ascoli
space. Extending one of the main results of [18] and [16] we show that $C_p(X)$
is sequentially Ascoli iff $X$ has the property $(\kappa)$. We give a necessary
condition on $X$ for which the space $C_k(X)$ is sequentially Ascoli. For every
metrizable abelian group $Y$, $Y$-Tychonoff space $X$, and nonzero countable
ordinal $\alpha$, the space $B_\alpha(X,Y)$ of Baire-$\alpha$ functions from
$X$ to $Y$ is $\kappa$-Fr\'{e}chet--Urysohn and hence Ascoli.

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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: supr-con
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
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The categories of this paper are: comp-gas
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The categories of this paper are: q-bio