The extended real number line is sometimes written as $\mathbb{R} \cup \{+\infty\} \cup \{-\infty\}$.
In ZFC, this would imply that $+\infty$ is some set. In other words, the string $+\infty$ is an identifier for a particular set.
So... which set is this? Or this this technically an abuse of notation?
For the purpose of compactification you may use any object that is not a real number, take $+\infty=\{\mathbb R_+\}$ for example.