Connected subsets of set

I have to find all connected subsets of {1/n, n$\in \mathbb{N} $ } $\cup $ {0}. I know all singletons of form {1/n} are connected. Is {0} connected?

I think it should be cause it cannot be split into 2 non empty seperated sets.

On any topological space, every singleton is a connected set. So, yes, $\{0\}$ is connected. And no subset of that space with more than one point is connected, because if $S\subset\{0\}\cup\left\{\frac1n\,\middle|\,n\in\Bbb N\right\}$ has more than one pont, then one such point is $\frac1n$, for some $n\in\Bbb N$, and if you take $\delta>0$ so small that $\left[\frac1n-\delta,\frac1n+\delta\right]\cap S=\left\{\frac1n\right\}$, then $S\cap\left(\frac1n-\delta,\frac1n+\delta\right)$ is both closed and open in $\{0\}\cup\left\{\frac1n\,\middle|\,n\in\Bbb N\right\}$.