  Being motivated by the famous Kaplansky theorem we study various sequential
properties of a Banach space $E$ and its closed unit ball $B$, both endowed
with the weak topology of $E$. We show that $B$ has the Pytkeev property if and
only if $E$ in the norm topology contains no isomorphic copy of $\ell_1$, while
$E$ has the Pytkeev property if and only if it is finite-dimensional. We extend
Schl\"uchtermann and Wheeler's result by showing that $B$ is a (separable)
metrizable space if and only if it has countable $cs^\ast$-character and is a
$k$-space. As a corollary we obtain that $B$ is Polish if and only if it has
countable $cs^\ast$-character and is \v{C}ech-complete, that supplements a
result of Edgar and Wheeler.
