Let $X$ be a Tychonoff topological space. $Y\subset X$ is $\omega$-dense in $X$, i.e., every point of $X$ is in the closure of some countable subset of $Y$, and $D=X\setminus Y$ is closed discrete in $X$. Then is the space $X$ discretely generated (see this link)?
( A space is called discretely generated if for every $A\subset X$ with $x \in cl(A)$ there is a discrete $D\subset A$ such that $x \in cl(D)$ ).
Thanks very much.