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

Kurilić, Miloš S., and Pavlović, Aleksandar. "A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube." Czechoslovak Mathematical Journal 64.2 (2014): 519-537. <http://eudml.org/doc/262025>.

@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 -