Connectedness of natural numbers

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?)

• Give it only two open sets $\varnothing$ and $\mathbb{N}$. There are other more interesting examples too. – Randall Oct 20 '20 at 16:45
• Yes, I suppose I meant more interesting than just that trivial example. – Prasiortle Oct 20 '20 at 16:47

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