On convergent sequences and copies of \(\beta N\) in the Stone space of one Boolean algebra (Q2850648)

Let \(M_0=\emptyset\) and \(M_n=\{0, \ldots, n-1\}\) for every natural \(n>0\). Given a number \(n\in \omega\), consider the set \(P_n=\{f:\) \(f\) is a function from \(M_n\) to \(\omega\) such that \( f(i)\leq i+1\) for any \(i\in M_n\}\) and let \(N=\bigcup\{P_n: n\in \omega\}\). Denote by \(T\) the set \(\{\pi\in N^\omega: \text{dom}(\pi(n))=M_{n+1}\) for all \(n\in \omega\}\) and let \(C_s=\{t\in N: t|\text{dom}(s)=s\}\) for each \(s\in N\). We will also need the set \(C_\pi=\bigcup\{C_{\pi(n)}: n\in \omega\}\) for every \(\pi\in T\). The letter \(B\) will be used for the Boolean algebra generated by the family \(B'=\{C_\pi: \pi\in T\}\) in the set \(\{X: X\subseteq N\}\). Observe that \(\{\{s\}: s\in N\}\cup\{C_s: s\in N\}\subseteq B\). Given \(s,t\in N\) say that \(s\leq t\) if \(t\) is an extension of \(s\); a set \(A\subseteq N\) is called antichain if it consists of incomparable elements of \(N\). A set \(C\subseteq N\) is a chain if any two elements of \(B\) are comparable.NEWLINENEWLINEIf \(BN\) is the Stone space of the Boolean algebra \(B\), then each \(s\in N\) can be identified with the ultrafilter \(\xi_s\) with \(\{s\}\in \xi_s\) so \(BN\) is a compactification of the countable discrete space \(N\). It was introduced by \textit{M. G. Bell} in his paper [Topol. Proc. 5, 11--25 (1981; Zbl 0464.54017)]. Bell proved that \(BN\) has a ccc non-separable remainder.NEWLINENEWLINEThe author of the paper under review continues the study of the space \(BN\). He proves that, for any set \(A\subseteq N\) such that \(\overline A \setminus A\) is a singleton, the set \(A\setminus K\) is a chain for some finite \(K\subseteq A\). Another result states that if \(A\subseteq N\) and \(\overline A\) is homeomorphic to \(\beta N\), then \(A\) is the union of finitely many antichains. It is also shown that there exist two antichains \(A,E\subseteq N\) such that both spaces \(\overline A\) and \(\overline E\) are homeomorphic to \(\beta N\) while the space \(\overline{A\cup E}\) is not homeomorphic to \(\beta N\).