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Is there any topology (or metric) on $\mathbb{N}$ that makes it connected (other than, trivially, the indiscrete topology)? (Clearly under the usual discrete topology the natural numbers are disconnected, but what about others?)

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    $\begingroup$ Give it only two open sets $\varnothing$ and $\mathbb{N}$. There are other more interesting examples too. $\endgroup$ – Randall Oct 20 '20 at 16:45
  • $\begingroup$ Yes, I suppose I meant more interesting than just that trivial example. $\endgroup$ – Prasiortle Oct 20 '20 at 16:47
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Consider the topology whose base is $\{ a + b\mathbb{N_0} : (a, b) = 1\}$, where $\mathbb{N}_0 = \mathbb{N} \cup \{0\}$. This is called the Golomb space. It is even Hausdorff.

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Consider the topology whose open sets are $\emptyset$ and all the subsets containing $1$.

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  • $\begingroup$ Yes! These "particular point" spaces can give some really interesting/simple counterexamples. $\endgroup$ – Randall Oct 20 '20 at 16:53

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