A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube

@article{Kurilić2014,
abstract = {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.},
author = {Kurilić, Miloš S., Pavlović, Aleksandar},
journal = {Czechoslovak Mathematical Journal},
keywords = {complete Boolean algebra; convergence structure; algebraic convergence; forcing; Cantor cube; Aleksandrov cube; small cardinal; complete Boolean algebras; sequential convergence; Cantor cube; Aleksandrov cube; forcing; adding reals},
language = {eng},
number = {2},
pages = {519-537},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube},
url = {http://eudml.org/doc/262025},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Kurilić, Miloš S.
AU - Pavlović, Aleksandar
TI - A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 519
EP - 537
AB - 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.
LA - eng
KW - complete Boolean algebra; convergence structure; algebraic convergence; forcing; Cantor cube; Aleksandrov cube; small cardinal; complete Boolean algebras; sequential convergence; Cantor cube; Aleksandrov cube; forcing; adding reals
UR - http://eudml.org/doc/262025
ER -