Let $a,b \in \mathbb{Z}, b \neq 0$ and $S(a,b):= \{a + nb \in \mathbb{Z}: n\in \mathbb{Z} \} \subseteq \mathbb{Z}$. We define the Furstenberg's topology as following: \begin{align*} W \subseteq \mathbb{Z} \, \, \, \text{open iff} \, \, \, W = \emptyset \, \, \text{or} \, \, \forall \, a\in W \, \, \exists \, b \neq 0 \, \text{such that} \, \, S(a,b) \subseteq W \end{align*} I have to show that $\mathbb{Z}$ with this topology is not compact.

I showed that is not connected but I can't see how to do with the compactness.

Any suggestions? Thanks in advance!

  • $\begingroup$ This topology should not be credited to Furstenberg. It is the pro-goup topology on $\mathbb{Z}$, a much older definition. $\endgroup$ – J.-E. Pin Apr 18 '18 at 9:22

Consider these subsets of $\mathbb Z$:

  • the odd numbers (numbers of the form $2a+1$);
  • the multiples of $3$ (numbers of the form $3a$);
  • even numbers which are not multiples of $4$ (numbers of the form $4a+2$);
  • multiples of $4$ which are not multiples of $8$ (numbers of the form $8a+4$)

and so on. They form an open cover of $\mathbb Z$, but this open cover has no finite subcover. In fact, no finite number of elements of this open cover contain every power of $2$.

  • $\begingroup$ How about the 0? Is possible that is not contained in no one of these three sets? $\endgroup$ – userr777 Apr 18 '18 at 10:46
  • $\begingroup$ @userr777 I've edited my answer. I hope that everything is correct now. $\endgroup$ – José Carlos Santos Apr 18 '18 at 10:59

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