Understanding the theorem 1.1 of a paper of D. Basile and A. Bella The paper is here.
Given a collection of sets $\mathcal V$ and a set $A$,  $\mathcal V[A]=\{V \in \mathcal V: V\cap A\not= \emptyset\}.$ 
The weak extent of a topological space $X$, denoted by $we(X)$, is the smallest cardinal $\kappa$ such that for any open cover $\mathcal U$ of $X$ there exists a set $A\subseteq X$ such that $|A|\le \kappa$ and $\mathcal U[A]$ is  cover of $X$.
A cover $\mathcal V$ of  a set $X$ is separating if $\bigcap \mathcal V[x]=\{x\}$ for every $x\in X$.
The point seperating weight of a topological space $X$, denoted by $psw(X)$, is the smallest cardinal $\kappa$ such that there eists a separating open cover $\mathcal V$ of $X$ such that $|\mathcal V[x]|\le \kappa$ for every $x \in X$.

Theorem 1.1 If $X$ is a $T_1$ space, then $|X|\le we(X)^{psw(X)}$.

I don't know that how the authors construct the set $S_w$ for any $W \subseteq \mathcal V_\alpha$.
Could you help me?
 A: If I understand your question correctly, you're asking about this part:

for any $\mathcal W\subseteq \mathcal V_\alpha$ satisfying $\bigcap\mathcal W\ne\emptyset$, choose a non-empty set $S_{\mathcal W}\subseteq \bigcap\mathcal W$ in such a way that, for some $\mathcal W^*\supseteq\mathcal W$, the family $\{\mathcal V[x] : x\in S_\mathcal W\}$ is maximal with respect to the property that $\mathcal V[p]\cap\mathcal V[q]$ for distinct $p, q \in S_{\mathcal W}$.

Since $\bigcap\mathcal W\ne\emptyset$, we can choose some $x_0\in\bigcap\mathcal W$ and set $\mathcal W^*=\mathcal V[x_0]$. It is easy to see that $\mathcal W^*\supseteq \mathcal W$ directly from the definition of $\mathcal V[x_0]$. (All elements of $\mathcal W$ are sets from $\mathcal V$ that contain the point $x_0$.) 
We already have one set $\mathcal V[x_0]$ and we add the sets of the form $\mathcal V[p]$ until no more sets can be added. (More formally you do this by Zorn's Lemma applied to the systems of sets having the form $\{\mathcal V[x] : x\in S\}$ and fulfilling the property $\mathcal V[p]\cap\mathcal V[q]=\mathcal W^*$ for $p\ne q$, $p,q\in S$.) 

Moreover, if there are distinct $p, q \in X$  satisfying $\mathcal V[p] \cap \mathcal V[q] =\mathcal W$, we will choose $S_{\mathcal W}$ in such a way that $\mathcal W^* = \mathcal W$ and $\{p, q\} \subseteq\mathcal S_{\mathcal W}$. 

In this case the construction is basically the same, but you don't start with a single point $x_0$, but with two points $p$, $q$ instead.
