A space $X$ is said to be star-Lindelöf if for every open cover $\mathcal U$ of $X$ there exists a countable subset $\mathcal V$ of $\mathcal U$ such that $\operatorname{St}(\bigcup\mathcal V,\mathcal U)=X$.

A space $X$ has

**discrete countable chain condition**(DCCC) if every discrete family of nonempty open sets is countable.

Does there exist a star-Lindelöf space which is not DCCC?

discretefamilies of open sets have cardinality at most one. $\endgroup$