# How could I see the space $Y$ is closed?

In the picture below:

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?

• When saying "the paper" you should probably add a citation. – Asaf Karagila Feb 25 '13 at 11:20
• Recall, in part, this past question of yours asking about why sequential spaces are countably tight. – user642796 Feb 25 '13 at 11:25
• 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. – user642796 Feb 25 '13 at 11:29

$\newcommand{\cl}{\operatorname{cl}}$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 $x\in\cl A\subseteq\cl F_\beta=F\beta\subseteq Y$.