The k_(R)-property of free Abelian topological groups and products of sequential fans

A space $X$ is called a $k_{R}$-space, if $X$ is Tychonoff and the necessary and sufficient condition for a real-valued function $f$ on $X$ to be continuous is that the restriction of $f$ to each compact subset is continuous. In this paper, we discuss the $k_{R}$-property of products of sequential fans and free Abelian topological groups by applying the $\kappa$-fan introduced by Banakh. In particular, we prove the following two results: (1) The space $S_{\omega_{1}}\times S_{\omega_{1}}$ is not a $k_{R}$-space. (2) The space $S_{\omega}\times S_{\omega_{1}}$ is a $k_{R}$-space if and only if $S_{\omega}\times S_{\omega_{1}}$ is a $k$-space if and only if $\mathfrak b>\omega_1$. These results generalize some well-known results on sequential fans. Furthermore, we generalize some results of Yamada on the free Abelian topological groups by applying the above results. Finally, we pose some open questions about the $k_{R}$-spaces.