I am self studying topology from Wayne Patty and I have a question in Theorem 2.48 on page 90. Subsection is Weak topology and product topology.
The question is that I am not able to understand how $\mathscr B$ is countable in 4th and 5th line of the proof. It is clear to me that for all $\alpha \in \Lambda- \Gamma$, $\pi_{\alpha}^{-1} (B_{\alpha}) =X$. But still I am not able to deduce how $\mathscr B$ is countable.
Please help me with that!
(You may already have realized this, but note that there is a typo in the fourth line: it should read ‘where $\alpha\in\Lambda$ and $B_\alpha\in\mathscr{B}_\alpha$’.)
For each $\alpha\in\Lambda$ let
$$\mathscr{S}_\alpha=\left\{\pi_\alpha^{-1}[B]:B\in\mathscr{B}_\alpha\right\}\,,$$
and let
$$\mathscr{S}=\bigcup_{\alpha\in\Lambda}\mathscr{S}_\alpha\,.$$
If $\alpha\in\Lambda\setminus\Gamma$, then $\mathscr{B}_\alpha=\{X_\alpha\}$, so $\mathscr{S}_\alpha=\left\{\pi_\alpha^{-1}[X_\alpha]\right\}=\{X\}$. If $\alpha\in\Gamma$, then $\mathscr{B}_\alpha$ is countable, so $\mathscr{S}_\alpha$ is also countable. Thus,
$$\mathscr{S}=\{X\}\cup\bigcup_{\alpha\in\Gamma}\mathscr{S}_\alpha$$
is the union of countably many countable sets and is therefore countable. Finally, Patty’s $\mathscr{B}$ is the set of all intersections of finite subsets of $\mathscr{S}$. $\mathscr{S}$, being countable, has only countably many finite subsets, so $\mathscr{B}$ is also countable.