Stone spaces of some Boolean algebras

R. A. Golovastov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 45 (2015) no. 1, pp. 3-36

Résumé

We study the Stone spaces of some Boolean algebras and establish relations between subsets of this spaces and Chech–Stone space $\beta\omega$, Cantor set, and other spaces. We consider three countable partially ordered sets and two type of Boolean algebras for each set. First, we consider space $S\mathfrak B_{1,1}$ constructered by M. Bell. We prove existence of subsets homeomorphic to $\beta\omega$ and convergent sequences in $S\mathfrak B_{1,1}$. For space $S\mathfrak B_{1,2}$, we prove that there are clopen subsets which is homeomorphic to $\beta\omega$ and remainder $S\mathfrak B_{1,2}^*$ consists of isolated points. We describe clopen subsets of $S\mathfrak B_{1,1}$ which are gomeomorphic to $\beta\omega$. We construct two examples: subset of $\mathfrak{N}_2$ which closure is non-open copy of $\beta\omega$ and subset of $\mathfrak{N}_2$ which closure is clopen and not gomeomorphic to $\beta\omega$. $S\mathfrak B_{1,2}$ is closure subset of $S\mathfrak B_{1,1}$ and $S\mathfrak B_{1,2}^*$ is nowhere dense in $S\mathfrak B_{1,1}^*$. Next, we consider the space $S\mathfrak B_{1,3}$. The subspace of free ultrafilters of $S\mathfrak B_{1,3}$ has the countable Suslin number, but is not separable. The points of the space are described as ultrafilters possessing basis of certain types. Next, we consider the spaces $S\mathfrak B_{2,1}$, $S\mathfrak B_{2,2}$, and $S\mathfrak B_{2,3}$. Boolean algebras for those Stone spaces have more simple structure. $S\mathfrak B_{2,3}$ is homeomorphic to Cantor set. The subset of free ultrafilters $S\mathfrak B_{2,3}^*$ is homeomorphic to the set of irrational numbers with natural topology. The subsets of free ultrafilters $S\mathfrak B_{1,3}^*$ and $S\mathfrak B_{1,3}^*$ are homeomorphic to Cantor set.

Détail

Mots-clés : compactification
Keywords: Boolean algebra, Stone space, ultrafilter.