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 Izv. IMI UdGU, 2015, Issue 1(45), Pages 3–36 (Mi iimi293)

Stone spaces of some Boolean algebras

R. A. Golovastov

Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Abstract: 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.

Keywords: compactification, Boolean algebra, Stone space, ultrafilter.

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UDC: 515.122.536
MSC: 54D35

Citation: R. A. Golovastov, “Stone spaces of some Boolean algebras”, Izv. IMI UdGU, 2015, no. 1(45), 3–36

Citation in format AMSBIB
\Bibitem{Gol15} \by R.~A.~Golovastov \paper Stone spaces of some Boolean algebras \jour Izv. IMI UdGU \yr 2015 \issue 1(45) \pages 3--36 \mathnet{http://mi.mathnet.ru/iimi293} \elib{https://elibrary.ru/item.asp?id=23754964}