Stone–Čech Compactification of $\mathbb{Z}$ with Fürstenberg Topology

We can also describe $\beta(\mathbb{Z},\mathcal{T})$ in terms of ultrafilters on Boolean algebras. I claim that $\beta(\mathbb{Z},\mathcal{T})$ is the space of ultrafilters on the Boolean algebra of clopen sets in $(\mathbb{Z},\mathcal{T})$ where $\mathcal{T}$ is the Fürstenberg topology.

Recall that a space $X$ is zero-dimensional if it has a basis of clopen sets, and recall that a zero set on a space $X$ is a set of the form $f^{-1}(0)$ for some continuous $f:X\rightarrow\mathbb{R}$. A completely regular space $X$ is said to be strongly zero-dimensional if the Stone-Čech compactification $\beta X$ is zero-dimensional. It can be shown that a completely regular space $X$ is strongly zero-dimensional if and only if whenever $Z_{1},Z_{2}\subseteq X$ are disjoint zero sets, there is a clopen set $C\subseteq X$ with $Z_{1}\subseteq C,Z_{2}\subseteq C^{c}$ [1 p. 85]. In other words, a completely regular space is strongly zero-dimensional iff every pair of zero sets is separated by a clopen set. If $X$ is zero-dimensional, then let $\mathfrak{B}(X)$ denote the Boolean algebra of clopen subsets of $X$ and let $\zeta X$ be the space of ultrafilters on $\mathfrak{B}(X)$. Then $\zeta X$ is in a sense the maximal zero-dimensional compactification of $X$ which is called the Banaschewski compactification. If $X$ is strongly zero-dimensional, then the Banaschewski compactification $\zeta X$ is precisely the Stone-Čech compactification. In [1. p. 86] it states that zero-dimensionality and strong zero-dimensionality are equivalent in Lindelöf spaces. Therefore since $(\mathbb{Z},\mathcal{T})$ is zero-dimensional and Lindelöf, the space $(\mathbb{Z},\mathcal{T})$ is strongly zero-dimensional. We conclude that $\beta(\mathbb{Z},\mathcal{T})=\zeta(\mathbb{Z},\mathcal{T})$ is the space of ultrafilters on $\mathfrak{B}(\mathbb{Z},\mathcal{T})$.

In order to clear up some confusion about the space $(\mathbb{Z},\mathcal{T})$ and its Stone-Čech compactification, I will outline some basic facts about $(\mathbb{Z},\mathcal{T})$ and $\beta(\mathbb{Z},\mathcal{T})$.

I claim that the space $(\mathbb{Z},\mathcal{T})$ has an infinite partition into clopen sets. It is not too hard to give an explicit example of such a partition. For a more slick proof, assume that $(\mathbb{Z},\mathcal{T})$ has no partition into countably many clopen sets. If $\mathcal{U}$ is an open cover of $\mathbb{Z}$, then there is a clopen cover $\{C_{n}|n\in\mathbb{N}\}$ that refines $\mathcal{U}$. If we set $D_{n}=C_{n}\setminus(C_{0}\cup...\cup C_{n-1})$ for all $n$, then $\{D_{n}|n\in\mathbb{N}\}$ is a partition of $(\mathbb{Z},\mathcal{T})$ into finitely many clopen sets that refines $\mathcal{U}$, so $(\mathbb{Z},\mathcal{T})$ is compact. This is a contradiction. Therefore $(\mathbb{Z},\mathcal{T})$ has a partition into countably many clopen sets.

In particular, there is a continuous surjective function $f:(\mathbb{Z},\mathcal{T})\rightarrow\mathbb{N}$ where $\mathbb{N}$ has the discrete topology. Therefore the map $f$ extends to a continuous surjective function $\bar{f}:\beta(\mathbb{Z},\mathcal{T})\rightarrow\beta\mathbb{N}$. Since $|\beta\mathbb{N}|=2^{\mathbb{c}}$, we conclude that $|\beta(\mathbb{Z},\mathcal{T})|=2^{\mathbb{c}}$ as well. We conclude that the Stone-Cech compactification $\beta(\mathbb{Z},\mathcal{T})$ is much larger than the pro-finite completion of $\mathbb{Z}$.

[1] The Stone-Čech Compactification, Russell Walker (1970)

Topologically speaking $\mathbb{Z}$ with the topology mentioned above is just (homeomorphic to) the space of rational numbers. The space $\beta\mathbb{Q}$ has been studied a lot (not as much as $\beta\mathbb{N}$), for example by Eric van Douwen in Remote Points (MR link), freely available here at DMLPL.