Question: Show that $\mathbb{R}$ is a closed subset of $\mathbb{R}$.
$\mathbb{R}\setminus \mathbb{R}=\left \{ x \in \mathbb{R} \mid x\notin\mathbb{R} \right \}.$
I need to show that $\forall x \in \mathbb{R}\setminus \mathbb{R}, \exists \epsilon >0$ s.t $B_{\epsilon }\left ( x \right )\subseteq \mathbb{R}\setminus \mathbb{R}$.
But, the complement of $\mathbb{R}$ itself doesn't make sense. Or rather, for it to make sense, it must be an empty set.