The left, the right and the sequential topology on Boolean algebras
Abstract
For the algebraic convergence $\lambda_{\mathrm{s}}$,
which generates the well known sequential topology $\tau_s$ on a complete Boolean algebra ${\mathbb B}$,
we have $\lambda_{\mathrm{s}}=\lambda_{\mathrm{ls}}\cap \lambda_{\mathrm{li}}$, where the convergences
$\lambda_{\mathrm{ls}}$ and $\lambda_{\mathrm{li}}$ are defined by
$\lambda_{\mathrm{ls}}(x)=\{ \limsup x\}\!\upar$ and $\lambda_{\mathrm{li}}(x)=\{ \liminf x\}\downar$
(generalizing the convergence of sequences on the Alexandrov cube and its dual).
We consider the minimal topology $\mathcal{O}_{\mathrm{lsi}}$ extending
the (unique) sequential topologies $\mathcal{O}_{\lambda_{\mathrm{ls}}}$ (left) and $\mathcal{O}_{\lambda_{\mathrm{li}}}$ (right)
generated by the convergences $\lambda_{\mathrm{ls}}$ and $\lambda_{\mathrm{li}}$
and establish a general hierarchy between all these topologies and the corresponding a priori and a posteriori convergences.
In addition, we observe some special classes of algebras and, in particular, show that
in $(\omega,2)$-distributive algebras we have $\lim_{{\mathcal O}_{\mathrm{lsi}}}=\lim_{\tau _{\mathrm{s}} }=\lambda _{\mathrm{s}}$,
while the equality $\mathcal{O}_{\mathrm{lsi}}=\tau_s$ holds in all Maharam algebras. On the other hand,
in some collapsing algebras we have a maximal (possible) diversity.
%\\[1mm]
which generates the well known sequential topology $\tau_s$ on a complete Boolean algebra ${\mathbb B}$,
we have $\lambda_{\mathrm{s}}=\lambda_{\mathrm{ls}}\cap \lambda_{\mathrm{li}}$, where the convergences
$\lambda_{\mathrm{ls}}$ and $\lambda_{\mathrm{li}}$ are defined by
$\lambda_{\mathrm{ls}}(x)=\{ \limsup x\}\!\upar$ and $\lambda_{\mathrm{li}}(x)=\{ \liminf x\}\downar$
(generalizing the convergence of sequences on the Alexandrov cube and its dual).
We consider the minimal topology $\mathcal{O}_{\mathrm{lsi}}$ extending
the (unique) sequential topologies $\mathcal{O}_{\lambda_{\mathrm{ls}}}$ (left) and $\mathcal{O}_{\lambda_{\mathrm{li}}}$ (right)
generated by the convergences $\lambda_{\mathrm{ls}}$ and $\lambda_{\mathrm{li}}$
and establish a general hierarchy between all these topologies and the corresponding a priori and a posteriori convergences.
In addition, we observe some special classes of algebras and, in particular, show that
in $(\omega,2)$-distributive algebras we have $\lim_{{\mathcal O}_{\mathrm{lsi}}}=\lim_{\tau _{\mathrm{s}} }=\lambda _{\mathrm{s}}$,
while the equality $\mathcal{O}_{\mathrm{lsi}}=\tau_s$ holds in all Maharam algebras. On the other hand,
in some collapsing algebras we have a maximal (possible) diversity.
%\\[1mm]
Refbacks
- There are currently no refbacks.