Product of moscow spaces 
Let $\{X_a : a\in A\}$ be a family of topological spaces such that $X_K=\prod\limits_{a\in K}X_a$ is a Moscow space of countable $o$-tightness, for every finite subset $K$ of $A$. 
  Then the product space $X =\prod\limits_{a\in A}X_a$ is Moscow. 

The $o$-tightness of a space $X$ is said to
be countable if whenever a point $a$ belongs
to the closure of $\bigcup\gamma$
, where $\gamma$
 is any family of open sets in
$X$, then there exists a countable subfamily $\eta$ of $\gamma$ 
 such that $a$ is
in the closure of $\bigcup\eta$.


*

*Referense : A. Arhangel'skii‎, Moscow spaces and topological groups, Topology
Pro. 25 (2000), P402‎.

*A. Arhangel'skii‎, Topological Groups and Related Structures, Theorem 6.3.12.

I spent a lot of time to understand the proof of the theorem and I have a few questions, Which I can not find their answers. I am looking for a proof that is more detailed than the reference.
Thanks.
 A: $\newcommand{\cl}{\operatorname{cl}}$I use the notation and terminology of the references. As in the paper, let $U$ be any open set in $X$, let $x\in\cl_XU$, let $\gamma$ be the set of open $\omega$-cubes with finite core that are contained in $U$, and let $A_0$ be any non-empty countable subset of $A$.
Suppose that a non-empty countable $A_n\subseteq A$ has been defined. Let $K=A_n$. By Lemma $2.23$ $ot(X_K)\le\omega$, so there is a countable $\gamma_n\subseteq\gamma$ such that 
$$p_K(x)\in\cl_{X_K}\bigcup_{V\in\gamma_n}p_K[V]\;.$$
By Lemma $2.22$ the space $X_K$ is Moscow, so there must be a $G_\delta$-set $F_n$ in $X_K$ such that 
$$p_K(x)\in F_n\subseteq\cl_{X_K}\bigcup_{V\in\gamma_n}p_K[V]\;;$$
let $P_n=p_K^{-1}[F_n]$, and note that $P_n$ is a $G_\delta$-set in $X$, and $x\in P_n$. Now let $$A_{n+1}=A_n\cup\bigcup_{V\in\gamma_n}A_V\;,$$ where $A_V$ is the core of $V$, to complete the $n$-th step of the recursive construction.
Let $P=\bigcap_{n\in\omega}P_n$; clearly $P$ is a $G_\delta$-set in $X$ containing $x$, so to complete the proof we need only show that $P\subseteq\cl_XU$. Let $H=\cl_X\bigcup\eta\subseteq\cl_XU$, and let $y\in P$ be arbitrary; we’ll show that $y\in H$.
Let $M=\bigcup_{n\in\omega}A_n$ and $\eta=\bigcup_{n\in\omega}\gamma_n$. Every member of $\eta$ has its core in $M$, so for all $y\in X$ we have $y\in H$ iff $p_M(y)\in p_M[H]$. Let $W$ be any standard basic open nbhd of $p_M(y)$ in the product $X_M$; i.e, $p_M(y)\in W\subseteq X_M$, and $W$ is an open $\omega$-cube with finite core $F\subseteq M$. The sets $A_n$ are increasing with $n$, and $F$ is finite, so $F\subseteq A_n$ for some $n\in\omega$. Since $p_{A_n}(y)\in F_n$, it follows from the construction that 
$$p_{A_n}[W]\cap\bigcup_{V\in\gamma_n}p_{A_n}[V]\ne\varnothing$$
and hence that 
$$W\cap\bigcup_{V\in\eta}p_M[V]\supseteq W\cap\bigcup_{V\in\gamma_n}p_M[V]\ne\varnothing\;.$$
$W$ was arbitrary, so $p_M(y)\in p_M[H]$, and hence $y\in H$, as desired. $\dashv$
