Geodesic
A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 519-537
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We compare the forcing-related properties of a complete Boolean algebra ${\mathbb B}$ with the properties of the convergences $\lambda _{\mathrm s}$ (the algebraic convergence) and $\lambda _{\mathrm {ls}}$ on ${\mathbb B}$ generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that $\lambda _{\mathrm {ls}}$ is a topological convergence iff forcing by ${\mathbb B}$ does not produce new reals and that $\lambda _{\mathrm {ls}}$ is weakly topological if ${\mathbb B}$ satisfies condition $(\hbar )$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $\lambda _{\mathrm {ls}}$ is a weakly topological convergence, then ${\mathbb B}$ is a $2^{\mathfrak h}$-cc algebra or in some generic extension the distributivity number of the ground model is greater than or equal to the tower number of the extension. So, the statement “The convergence $\lambda _{\mathrm {ls}}$ on the collapsing algebra ${\mathbb B}=\mathop {\mathrm {ro}} (^{\omega }\omega _2)$ is weakly topological“ is independent of ZFC.
We compare the forcing-related properties of a complete Boolean algebra ${\mathbb B}$ with the properties of the convergences $\lambda _{\mathrm s}$ (the algebraic convergence) and $\lambda _{\mathrm {ls}}$ on ${\mathbb B}$ generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that $\lambda _{\mathrm {ls}}$ is a topological convergence iff forcing by ${\mathbb B}$ does not produce new reals and that $\lambda _{\mathrm {ls}}$ is weakly topological if ${\mathbb B}$ satisfies condition $(\hbar )$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $\lambda _{\mathrm {ls}}$ is a weakly topological convergence, then ${\mathbb B}$ is a $2^{\mathfrak h}$-cc algebra or in some generic extension the distributivity number of the ground model is greater than or equal to the tower number of the extension. So, the statement “The convergence $\lambda _{\mathrm {ls}}$ on the collapsing algebra ${\mathbb B}=\mathop {\mathrm {ro}} (^{\omega }\omega _2)$ is weakly topological“ is independent of ZFC.
DOI : 10.1007/s10587-014-0117-6
Classification : 03E17, 03E40, 06E10, 54A20, 54D55
Keywords: complete Boolean algebra; convergence structure; algebraic convergence; forcing; Cantor cube; Aleksandrov cube; small cardinal 
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     title = {A convergence on {Boolean} algebras generalizing the convergence on the {Aleksandrov} cube},
     journal = {Czechoslovak Mathematical Journal},
     pages = {519--537},
     year = {2014},
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Kurilić, Miloš S.; Pavlović, Aleksandar. A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 519-537. doi: 10.1007/s10587-014-0117-6

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