**Edit.** The comments and the other answers reveal that my proof has some gap. But I won't delete it, instead I've rewritten it as an *attempt* to prove $\beta \mathbb{Z} = \hat{\mathbb{Z}}$. I hope that the failure of this naive proof motivates to read the more sophisticated answers.

When $\mathbb{Z}$ is equipped with the Fürstenberg topology, do we have $\beta \mathbb{Z} = \hat{\mathbb{Z}}$?

The Fürstenberg topology is the subspace topology induced by the profinite completion $\hat{\mathbb{Z}} = \lim_{n>0} \mathbb{Z}/n$. The embedding $\mathbb{Z} \to \hat{\mathbb{Z}}$ is dense, hence for every compact Hausdorff space $X$ we get an injective map $\hom(\hat{\mathbb{Z}},X) \to \hom(\mathbb{Z},X)$. The question is whether it is surjective, because this would mean that $ \hat{\mathbb{Z}}$ satisfies the defining universal property of $\beta \mathbb{Z}$.

Let $f : \mathbb{Z} \to X$ be a continuous map. This means that for every $a \in \mathbb{Z}$, every open subset $U \subseteq X$ containing $f(a)$ already contains $f(a+n \mathbb{Z})$ for some $n>0$. Let $a=(a_1,a_2,\dotsc) \in \hat{\mathbb{Z}}$, i.e. $a_n \equiv a_m \bmod n$ for $n|m$. Since $X$ is compact, the net $(f(a_n))_{n>0}$ (using divisibility for the indices) has a convergent subnet, say $(f(a_{n(i)}))_{i \in I} \longrightarrow x$.

Actually any two subnets have the same limit: Assume that $(f(a_{m(j)}))_{j \in J} \longrightarrow y$. Choose open neighborhoods $U,V$ of $x,y$, it is enough to prove $U \cap V \neq \emptyset$ since $X$ is Hausdorff. For large $i$ we have that $f(a_{n(i)}) \in U$, and for large $j$ we have $f(a_{m(j)}) \in V$. Choose $b>0$ with $f(a_{n(i)} + b \mathbb{Z}) \subseteq U$ and $f(a_{m(j)} + b \mathbb{Z}) \subseteq V$. We may assume $n(i),m(j)|b$. For $p=n(i) m(j)$ we have $a_p \equiv a_{n(i)} \bmod n(i)$, hence $a_p \equiv a_{n(i)} \bmod b$. Similarily we get $a_p \equiv a_{m(j)} \bmod b$. Hence $f(a_p) \in U \cap V$.

Hence $\tilde{f}(a) := $(the limit of some subnet of $f(a_n)$) gives a well-defined map $\hat{\mathbb{Z}} \to X$. Clearly it agrees with $f$ on constant sequences. But now the problem seems to be that $\tilde{f}$ is not continuous ...

not the same– Yemon Choi Mar 27 '13 at 16:41