  The main purpose of this paper is to study \emph{$e$-separable spaces},
originally introduced by Kurepa as $K_0'$ spaces; we call a space $X$
$e$-separable iff $X$ has a dense set which is the union of countably many
closed discrete sets. We primarily focus on the behaviour of $e$-separable
spaces under products and the cardinal invariants that are naturally related to
$e$-separable spaces. Our main results show that the statement "there is a
product of at most $\mathfrak c$ many $e$-separable spaces that fails to be
$e$-separable'" is equiconsistent with the existence of a weakly compact
cardinal.
