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Let $X$ be a Banach space and $𝒮\mathrm{𝑒𝑞}\left(X^{**}\right)$ (resp., $X_{{\aleph }_0}$) the subset of elements $\psi \in X^{**}$ such that there exists a sequence ${\left(x_n\right)}_{n\ge 1}\subset X$ such that $x_n\to \psi$ in the $w^{*}$-topology of $X^{**}$ (resp., there exists a separable subspace $Y\subset X$ such that $\psi \in {\overline{Y}}^{w^{*}}$). Then: (i) if $Dens\left(X\right)\ge {\aleph }_1$, the property $X^{**}=X_{{\aleph }_0}$ (resp., $X^{**}=𝒮\mathrm{𝑒𝑞}\left(X^{**}\right)$) is ${\aleph }_1$-determined, i.e., $X$ has this property iff $Y$ has, for every subspace $Y\subset X$ with $Dens\left(Y\right)={\aleph }_1$; (ii) if $X^{**}=X_{{\aleph }_0}$, $(B\left(X^{**}\right),w^{*})$ has countable tightness; (iii) under the Martin’s axiom $\mathrm{𝑀𝐴}\left({\omega }_1\right)$ we have $X^{**}=𝒮\mathrm{𝑒𝑞}\left(X^{**}\right)$ iff $(B\left(X^{*}\right),w^{*})$ has countable tightness and $overline\text{co}\left(B\right)={\overline{\text{co}}}^{w^{*}}\left(K\right)$ for every subspace $Y\subset X$, every $w^{*}$-compact subset $K$ of $Y^{*}$, and every...