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In the picture below: enter image description here

which is the lemma 2.1 of the paper of Arhangelk'skii, I cannot understand the part that he tried the space $Y$ is closed. Could somebody help me?

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    $\begingroup$ When saying "the paper" you should probably add a citation. $\endgroup$ – Asaf Karagila Feb 25 '13 at 11:20
  • $\begingroup$ Recall, in part, this past question of yours asking about why sequential spaces are countably tight. $\endgroup$ – user642796 Feb 25 '13 at 11:25
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    $\begingroup$ The paper in question is A.V. Arhangel'skii and R.Z. Buzyakova, On linearly Lindelöf and strongly discretely Lindelöf spaces, Proc. Amer. Math. Soc. 127, pp.2449-2458, link. $\endgroup$ – user642796 Feb 25 '13 at 11:29
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$\newcommand{\cl}{\operatorname{cl}}$Suppose that $x\in\cl Y$; then there is a countable $A\subseteq Y$ such that $x\in\cl A$. For each $a\in A$ there is an $\alpha(a)<\tau$ such that $a\in F_\alpha(a)$. Let $\beta=\sup\{\alpha(a):a\in A\}$; then $\beta<\tau$, since the cofinality of $\tau$ is uncountable, and $A\subseteq F_\beta$, since the sets $F_\xi$ are nested. But then $x\in\cl A\subseteq\cl F_\beta=F\beta\subseteq Y$.

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