Let $E$ be a Banach space. Let us consider the following three sigma algebras on $E$.

$~~~~\mathcal{B}$= The $\sigma$-algebra coming from the norm topology.

$~~\mathcal{M}$= The sigma algebra coming from the weak topology $\sigma(E,E^*)$.

$\mathcal{M}_0$= The sigma algebra coming from the weakly basic nbhds.

It is easy to see that $\mathcal{B}=\mathcal{M}$ provided that $E$ is separable. By an extraordinary example, Fremlin proved the converse is no longer valid in general. What about the following implications?!

1- $E$ is separable if and only if $\mathcal{B}=\mathcal{M}_0$.

Even by a weaker one:

2- $E$ is separable if and only if $\mathcal{M}=\mathcal{M}_0$.