  A well-known result of R. Pol states that a Banach space $X$ has property
($\mathcal{C}$) of Corson if and only if every point in the weak*-closure of
any convex set $C \subseteq B_{X^*}$ is actually in the weak*-closure of a
countable subset of $C$. Nevertheless, it is an open problem whether this is in
turn equivalent to the countable tightness of $B_{X^*}$ with respect to the
weak*-topology. Frankiewicz, Plebanek and Ryll-Nardzewski provided an
affirmative answer under $\mathrm{MA}+\neg \mathrm{CH}$ for the class of
$\mathcal{C}(K)$-spaces. In this article we provide a partial extension of this
latter result by showing that under the Proper Forcing Axiom ($\mathrm{PFA}$)
the following conditions are equivalent for an arbitrary Banach space $X$:
  1) $X$ has property $\mathcal{E}'$;
  2) $X$ has weak*-sequential dual ball;
  3) $X$ has property ($\mathcal{C}$) of Corson;
  4) $(B_{X^*},w^\ast)$ has countable tightness.
  This provides a partial extension of a former result of Arhangel'skii. In
addition, we show that every Banach space with property $\mathcal{E}'$ has
weak*-convex block compact dual ball.
