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.
Not necessarily. Let $D$ be the $2$-point discrete space. It’s known that $D^{\mathfrak c}$ is not discretely generated. Let $X=D^{\mathfrak c}\sqcup(\omega+1)$, where $\omega$ has the order topology. Clearly $X$ is Tikhonov and not discretely generated. Let $Y=D^{\mathfrak c}\sqcup\omega$; $X\setminus Y=\{\omega\}$ is closed and discrete in $X$. Finally,, $Y$ is $\omega$-dense in $X$: if $y\in Y$, then $y\in\operatorname{cl}_X\{y\}$, and $\omega\in\operatorname{cl}_X(Y\setminus D^{\mathfrak c})$.
