The question is also posted here, however there is no answer.

Recently, I am reading the paper: *On linearly Lindelöf and strongly discretely Lindelöf spaces* by Arhangel'skii and Buzyakova. Here is the Lemma 2.2 in paper. (Sorry for the picture is not clear.)

The fifth line from last. How could I see that for any $a\in H$ and $z\in Z\setminus H$, there exists an element $V$ of $\mathcal{U}$ such that $a\in V$ and $z\notin V$? Thanks very much.