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Keywords:
$k_R$-spaces; $k$-spaces; $k$-networks; $\sigma $-hereditarily closure-preserving collections; point-countable collections
$k_R$-spaces; $k$-spaces; $k$-networks; $\sigma $-hereditarily closure-preserving collections; point-countable collections
Summary:
In this note we study the relation between $k_R$-spaces and $k$-spaces and prove that a $k_R$-space with a $\sigma $-hereditarily closure-preserving $k$-network consisting of compact subsets is a $k$-space, and that a $k_R$-space with a point-countable $k$-network consisting of compact subsets need not be a $k$-space.
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