A sequence \(\langle x_n\rangle_n\) in a complete Boolean algebra can be defined to converge to an element \(x\) in various ways. A very natural way is to define \(\lim x_n=x\) to mean that \(x\) is equal to both \(\limsup x_n\) and \(\liminf x_n\), this is called algebraic convergence. The authors observe that in case the algebra is a power set, \(\mathcal P(X)\), a sequence converges algebraically iff the sequence of characteristic functions converges in the Cantor cube \(2^X\). This leads them to consider convergence in the Aleksandrov cube, which is \(2^X\) with the product topology derived from the topology on \(\{0,1\}\), where \(\{0\}\) is the only non-trivial open set. This may be formulated solely in terms of Boolean algebras as: the sequence \(\langle x_n\rangle_n\) converges to \(x\) iff \(x\geq\limsup x_n\) (thus, limits are no longer unique).
A sequential convergence determines a topology: a set is closed if all limits of all sequences in the set belong to the set, and if convergence in the topology coincides with the given convergence, the latter is said to be topological. The main result of the paper states that the Aleksandrov-like convergence is topological iff the Boolean algebra adds no new reals; the same is true for Cantor-like convergence, [see M. S. Kurilić and A. Pavlović, Ann. Pure Appl. Logic 148, No. 1-3, 49-62 (2007; Zbl 1132.06008)]. One can enrich a convergence structure by stipulating that a sequence converges to a point iff every subsequence has a further subsequence that converges to the point in the original sense; if the new structure is topological then the original one is said to be weakly topological. The authors show that collapsing \(\omega_2\) to \(\omega\) with finite conditions may or may not be weakly topological.