I have a question about the proof of the Lemma 7.2 in the paper
I. Juhász, P. Koszmider and L. Soukup, A first countable, initially $\omega_{1}$-compact but non-compact space, Topology and its Applications 156 (2009), 1863-1879. doi:10.1016/j.topol.2009.04.004
I want to understand the argument in the highlighted part of the following proof:
Lemma 7.2. In $V [\mathcal{G}]$, for each set $A \in {\left[ \omega_{2} \times \mathbb{C} \right]}^{\omega_{1}}$ there is $\beta \in \omega_{2}$ such that $\left| A \cap U (\beta) \right| = \omega_{1}$.
Proof. Let $\dot{A}$ be a $P_{f}$-name for $A$ and assume that $p \in \mathcal{G}$ with $$ p \Vdash \dot{A} = \left\lbrace \dot{z}_{\xi} : \xi < \omega_{1} \right\rbrace \in {\left[ \omega_{2} \times \mathbb{C} \right]}^{\omega_{1}} . $$ We may assume that $p$ also forces that $\left\lbrace \dot{z}_{\xi} : \xi < \omega_{1} \right\rbrace$ is a one-one enumeration of $\dot{A}$. For each $\xi < \omega_{1}$ we may pick $p_{\xi} \leqslant p$ and $\alpha_{\xi} \in \omega_{2}$ with $\alpha_{\xi} \in a_{p_{\xi}}$ such that $p_{\xi} \Vdash \dot{z}_{\xi} = \left\langle \alpha_{\xi} , \dot{x}_{\xi} \right\rangle$. Let $\sup \left\lbrace \alpha_{\xi} : \xi < \omega_{1} \right\rbrace < \beta < \omega_{2}$. By Lemma 7.1 for each $\xi < \omega_{1}$ there is a condition $q_{\xi} \leqslant p_{\xi}$ such that $\alpha_{\xi} \in h_{q_{\xi}} (\beta , 0)$, hence $q_{\xi} \Vdash \dot{z}_{\xi} \in U (\beta)$. But $P_{f}$ satisfies CCC, so there is $q \in \mathcal{G}$ such that $q \Vdash \left| \{ \xi \in \omega_{1} : q_{\xi} \in \mathcal{G} \} \right| = \omega_{1}$. Clearly, then $q \Vdash | \dot{A} \cap U (\beta) | = \omega_{1}$. $\square$
