On topological properties of the weak topology of a Banach space

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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 ℓ₁, while E has the Pytkeev property if and only if it is finite-dimensional. We extend Schlüchtermann and Wheeler's result by showing that B is a (separable) metrizable space if and only if it has countable cs-character and is a k-space. As a corollary we obtain that B is Polish if and only if it has countable cs-character and is Čech-complete, that supplements a result of Edgar and Wheeler.