Some centered systems of sets and points defined by them

We study a compactification $BN$ of a countable discrete space $N$, which is constructed as the Stone space of a Boolean algebra, and consider the class of subsets from $BN$ that are copies of the space $\beta N$ (the Stone–Čech compactification of the space $N$). Characteristics of points of closure are obtained for centered systems of some families of subsets of the Boolean algebra.