An open neighbourhood assignment for a topological space $(X, \tau)$ is a map $U: X \to \tau$ such that $x \in U(x)$, for every $x \in X$. A space $X$ is called a $D$-space if for every open neighbourhood assignment $U$ there is a closed discrete set $D \subset X$ such that $X=\bigcup \{U(x): x \in D \}$.
The famous $D$-space problem of van Douwen asks whether every regular Lindelof space is a D-space.
Is every Banach space which is Lindelof in its weak topology a $D$-space?
I'm asking whether it is a $D$-space in its weak topology of course, because the strong topology is metrizable and it's easy to see that every metric space is a $D$-space.
