Reading an article I came across the following line, which botheres me since quite a while.
Let $E:=\bar{\mathbb{R}}^{d}_{0}:=([-\infty,0)\cup(0,+\infty])^d$ be the closure of $\mathbb{R}^d$ without ${0}$. Then - so the claim - a set $B\subseteq E$ is relatively compact if and only if it is bounded away from $0$. I mean obviously such a set $B$ doesn't have to be bounded above, hence it's closure cannot be bounded too. Why isn't that a problem? In $\mathbb{R^d}$ or $\mathbb{R^d}/\{0\}$ it most certainly would be one, so why and how does adding the "infinity"-points changes that?
It isn't a problem because $(a,\infty]$ is an open set for any $a\in \mathbb{R_{+}}$ - so you need only one open set to cover a "large" portion of the set. In particular, even if a set $B$ is unbounded, you can make it bounded by removing $B\setminus (1,\infty]$. What remains will now be bounded, but removing this set does not change the relative compactness of $B$.