I know R is closed and open but is it a closed and open interval? I know $\mathbb{R}$ is open and closed but is this the same as saying it's an open interval and a closed interval? 
 A: This really depends on your working definition of "interval" — that is, your textbook's definition or your teacher's.
If "interval" is defined as: 

  
*
  
*$I \subseteq \mathbb{R}$ is an interval $\iff$ there are $x, y \in \mathbb{R}$ such that $I$ is one of $[a,b], [a,b), (a,b]$ or $(a,b)$
  

then No, $\mathbb{R}$ is not an interval.
However, if "interval" is defined as: 


  
*$I \subseteq \mathbb{R}$ is an interval $\iff$ for all $a, b \in \mathbb{R}$ with $a \le b$, if $a, b \in I$ then $[a, b] \subseteq I$
  

then Yes, $\mathbb{R}$ is an interval.
NOTES


*

*According to 2., not just $\mathbb{R}$ itself but also all half-unbounded "rays" like $(-\infty, a)$ and $[b, +\infty)$ are intervals. According to 1., however, they are not.

*Definition 1. is more restrictive. BUT, if 1. is relaxed to allow $a, b \in \overline {\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, then the resulting definition is equivalent to 2.
In any case, $\mathbb{R}$ is not a closed interval, because if any "closed interval" contains its inf and sup; however, the "extended real $-\infty = \inf(\mathbb{R}) \notin \mathbb{R}$ (and similarly for $\sup$ and $+\infty$).
