Are $(\infty,\infty)$ and $(-\infty,-\infty)$ intervals? Are $(\infty,\infty)$ and $(-\infty,-\infty)$ intervals?
Also, is (1,1) an interval? if so, is it just an empty interval?
 A: Interval notation is exactly that: notation
It is a shorthand way of denoting certain types of subsets of $\mathbb{R}$.
$(a,\infty)$ for example is just an abbreviation for $\{x\in\mathbb{R}\,\vert\,x>a\}$
$(a,b)$ is shorthand for $\{x\in\mathbb{R}\,\vert\,a<x<b\}$
Given this understanding of the notation, constructions such as you ask about are not intervals. If you wish to call $(1,1)$ the empty set that is not particularly objectionable, but to call it an interval would be non-standard.
A: At least according to the standard definition given on Wikipedia, which is that an interval of reals is a convex set of reals (every real between any two members is also an member), $(1,1)$ is indeed the empty set and is an interval by definition.
$(\infty,\infty)$ can be and is readily interpreted in the affinely-extended reals in the same manner, giving an empty set as well. Same for $(-\infty,-\infty)$. However, notation wise they do not make sense if we insist on restricting to the real line alone.
[Two people now have downvoted my answer even though its first paragraph says essentially the same thing as Asaf Karagila's answer. Please read his answer properly as it explains quite well the basic logic needed to understand this "vacuous truth". Also, most people adopt such a definition that does not exclude empty intervals because it leads to more elegant theorems (see Noah Schweber's answer).]
A: To add to other answers:
$(a,b) = \{x \in \mathbb{R}: a < x < b\}$
$(a,b] = \{x \in \mathbb{R}: a < x \le b\}$
$[a,b) = \{x \in \mathbb{R}: a \le x < b\}$
$[a,b] = \{x \in \mathbb{R}: a \le x \le b\}$
And so forth. Similarly, $\infty$  or $-\infty$ as interval bounds are simply notation for no bounds on that end of the interval.
Also, an interval with one or no elements like $[a,a]$ or $(a,a)$ are often called degenerate intervals if you want to learn more about them.
