Is $ \{ \infty \} $ open or closed in $ \overline{ \mathbb R}$? Is $ \{ \infty \} $ open or closed in $ \overline{ \mathbb R}$ ?
Here  $\overline{ \mathbb R} = \mathbb R  \cup \{ \infty \} \cup \{ -\infty \} $.
 A: It's closed in the usual topology.
Your question seems to assume it's either open or closed, but most sets are neither open nor closed.
We want to define such things as $\lim\limits_{x\to3} f(x) = +\infty$.  That has to mean that for every open set $A$ containing $+\infty$, there is some open set $B$ containing $3$ small enough so that if $x\in B$ and $x\ne 3$, then $f(x)\in A$.  It is from such considerations as that that the definitions of open and closed subsets of $\overline{\mathbb{R}}$ are derived.
A set is closed precisely if its complement is open.  The set $(-a,a)$ is open.  The union of every set of open sets is open.  The union of all sets of the form $(-a,a)$ is all of $\mathbb{R}$, and clearly excludes both $+\infty$ and $-\infty$.  The set $\{-\infty\}\cup(-\infty,b)$ is open.  So we have
$$
\big(\{-\infty\}\cup(-\infty,b)\big)\cup\big( \bigcup_{a>0} (-a,a) \big)
$$
is open.  Its complement is $\{+\infty\}$.  So that set is closed.
A: Another way to look at it is to consider the complement $\{-\infty\} \cup \mathbb{R}$ which is a neighborhood of all of its elements, i.e. it is open.  Hence its complement, $\{\infty\}$, is closed.
