Origin of the $\bar{\mathbb{R}}$ notation I have the following exercise:
Show that closure of $\mathbb{R}$ is equal to $\mathbb{\mathbb{R}} \cup \{-\infty,+\infty\}$, justifying the notation $\bar{\mathbb{R}}$, for the order topology.
The definition of closure of a set $A$ that I have is it being the intersection of all closed sets containing A.
Now, if $\mathbb{R}$ is closed and $\mathbb{R} \subseteq \mathbb{R}$, why does the closure have to include $\{-\infty,+\infty\}$?
 A: You could regard it as a simple abuse of notation, but it is a motivated abuse of notation.
In the topological space $\mathbb R \cup \{-\infty,+\infty\}$ with the order topology, the closure of the subset $\mathbb R$ is equal to the whole space $\mathbb R \cup \{-\infty,+\infty\}$. So the following equation is simply true in that topological space:
$$\overline{\mathbb R} = \mathbb R \cup \{-\infty,+\infty\}
$$
So, now, which one would you rather write or even read? $\mathbb R \cup \{-\infty,+\infty\}$ or $\overline{\mathbb R}$? Those who like brevity of notation, even at the expense of abuse of notation, will use that equation as motivation (and "justification") for choosing $\overline{\mathbb R}$.
Let me also add that a topologist might go further down the "justification" path as follows: as in any topological space, it is obvious that the closure of $\mathbb R$ in itself is $\mathbb R$. So why waste good notation like $\overline{\mathbb R}$ on the closure of $\mathbb R$ in itself?
Most abuses of notation are motivated or justified in some such fashion, but that's a much longer story...
