About clopen set in $\mathbb R^n$

Question

Is there any clopen set in the metric space ($\mathbb R^n$,$d$) (where $d$ is the usual metric)

besides $\mathbb R^n$ & the empty set?

Answer 1

No. If there were such a set $U$, then setting $V = \mathbb{R}^n \setminus U$, $V$ will also be clopen, so that $\mathbb{R}^n = U \sqcup V$ is a decomposition of $\mathbb{R}^n$ as a union of nonempty disjoint open sets.

Answer 2

The metric spaces $X$ for which the only clopen subsets are $\emptyset$ and $X$ have a name: connected spaces. And, yes, $\mathbb{R}^n$ is connected. Therefore, it has no other clopen subsets.