May be this question turns out trivial, but I can't figure out. I asked on StackExchange, but no one answers.

Let $S$ be a set, or equivalently the topological space with discrete topology and $2$ two point set with discrete tiopology. The $βS$ be the Stone–Čech compactification of $S$. By Tychonoff theorem the topology on $2^S$ is compact with respect to the product topology. Note that $P(S)$ the powerset of $S$ is isomorphic of $2^S$ as a set. Let $(P(S),\tau_{prod})$ be topologycal space on $P(S)$ with topology induced by product topology on $2^S$.

Now consider $X=\{(A,\Sigma)|A\in \Sigma\}\subseteq P(S)\times \beta S$. Where ultrafilter $\Sigma\in \beta S$ varies in $\beta S$. All possible $(A,\Sigma)$ s.t. $A\in P(S)$, $\Sigma\in \beta S$ and $A\in \Sigma$

My question is can $X$ be a clopen subset if $S$ is infinite?

Thanks in advance.