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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.)

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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

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