Aleksandar Pavlović

The convergence of the sequences coding the ground model reals

We investigate the convergence λ1 on a complete Boolean algebra B defined in the following way: a sequence x = 〈xn : n ∈ ω〉 in B converges to the point lim sup x of B, if in each generic extension VB[G] the real coded by the name τx = {〈ň, xn〉 : n ∈ ω} belongs to the ground model V ; otherwise, x has no limit points. It is shown that λ1 generates the same topology as the convergence λ̄4, generalizing the sequential convergence on the Aleksandrov cube and that for a c.B.a. B the following conditions are equivalent: (1) The algebra B is (ω, 2)-distributive; (2) The (L2)-closure of λ1, λ̄1, is a topological convergence; (3) λ̄1 = λ̄4; (4) λ1 = λ4; and, for the algebras satisfying hcc(B) > c, (5) λ̄1 is a weakly topological convergence. Also, it is shown that the convergence λ̄1 is not weakly topological, if forcing by B produces splitting reals.