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.

We don't have to separate $0$ from the rest, instead, we have to show that if any subset of $K $ contains $0$ and another point, then that set is not connected. This shows that no point besides $0$ itself is in the same connected component at $0$.
