Some forcing related convergence structures on complete boolean algebras

Miloš S. Kurilić, Aleksandar Pavlović

Let convergences $\lambda_i:\Bbb B^{\omega}\to P(\Bbb{B})$, $i\leq 4$, on a complete Boolean algebra $Bbb{B}$ be defined in the following way. For a sequence $x=\langle x_n:n\in\omega\rangle$ in $\Bbb B$ and the corresponding $\Bbb B$-name for a subset of $\omega,\tau_x=\{\langle\tilde{n},x_n\rangle: n\in omega\}$, let $$ ambda_i(x)=\cases \{\|au_xext{ is infinite|\|\} & ext{if b_i(x)=1_{\Bbb B} \hfil\emptyset & ext{otherwise}, \endcases $$ where $b_1(x)=\|\tau(x)\text{ is finite or cofinite}\|$, $b_2(x)=\|\tau(x)\text{ is not unsupported}\|$, $b_3(x)=\|\tau(x)\text{ is not a splitting real\|$ and $b_4(x)=1_{\Bbb B}$. Then $\lambda_1$ is the algebraic convergence generating the sequential topology on $B$, while the convergences $\lambda_2$, $\lambda_3$ and $\lambda_4$, although different on each Boolean algebra producing splitting reals, generate the same topological convergence -- a generalization of the convergence on the Aleksandrov cube, considered in [18].