Which are the connected components of $K=\{1/n\mid n\in\mathbb{N}\}\cup\{0\}$? Which are the connected components of the topological subspace $K=\left\{\left.\dfrac{1}{n}\ \right|\ n\in\mathbb{N}\right\}\cup\{0\}$ of $\mathbb{R}$?
I think they are every single point, but I can't find open set of $\mathbb{R}$ separating the point $0$ from the others, thanks.

You cannot find such an open set. What you need to check is that all points of the form $\frac{1}{n}$ are isolated points.
Any subset $C$ with more than 1 point and at least one isolated point $p$ is disconnected (use $\{p\}$ and its complement in $C$, both of which are open and closed in $C$).
