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](http://dx.doi.org/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 $q \Vdash | \dot{A} \cap U (\beta) | = \omega\_{1}$. $\square$
> 
> 
> 



CCC Forcing and $\omega_1$ conditions
If there were no such $q$, then it would be forced, by all conditions, that $\{\xi\in\omega\_1:q\_\xi\in\mathcal G\}$ is countable and therefore bounded below $\omega\_1$. Consider a maximal antichain of conditions that force specific values ($\check\gamma$) for the bound. Since the antichain is countable, there are only countably many $\gamma$'s involved here, so there's a countable upper bound $\delta$ for all of them. But then, if $\delta<\xi<\omega\_1$, no condition can force $q\_\xi\in\mathcal G$, which is absurd as $q\_\xi$ forces this.


