# A question from Arhangel'skii-Buzyakova

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.

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If $a \in Y$ it is easy: then we can use one of members of $\gamma_a$ that misses $z$. If $a \in H \setminus Y$, it is in one of the $K_\alpha$, but then $z$ could be in all the members of the corresponding $\gamma_\alpha$: all we would get it that would not be in $X$ in that case (so the case $z \in X$ is also taken care of). I don't quite see how to handle $a \in H \setminus Y$ and $z \in Z \setminus X, z \notin H$. –  Henno Brandsma Mar 4 '13 at 21:40
@HennoBrandsma: yes. Maybe Something is wrong in this paper. –  Paul Mar 5 '13 at 3:49