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.