Are all open sets open intervals? Are all open sets open intervals? If not, provide an example. I think that all open intervals are open sets but not the other way around, but couldn't find an example.
 A: You haven't mentioned it, but I understand that you're talking about intervals on $\mathbb R$.
Are all open intervals open sets? Yes.
Open intervals on $\mathbb R$ are of the form $(a,b)$, where we let $a,b$ take on values in $[-\infty,\infty]$.  You can check that these are open sets.
Are all open sets open intervals? No!
The union of two open sets is open (more generally, any arbitrary union of open sets is open). So, take $(0,1)$ and $(2,3)$. These are open intervals, hence open sets. From what we just said, $(0,1) \cup (2,3)$ is also an open set, but not an open interval (let alone an open interval, this isn't even an interval).

P.S. What you should know:
When is $A\subset \Bbb R$ open?
$A\subset \Bbb R$ is open if for every $x\in A$, there exists some $\epsilon > 0$ so that $(x-\epsilon,x+\epsilon) \subset A$. In other words, for every point in $A$, you can find a neighborhood of that point which is contained in $A$.
A: This question ultimately depends on the set and topology. Since the tag is real-analysis, I take it your talking about $\mathbb{R}$ with the standard topology. In that case, you're correct that all open intervals are open sets, whereas not every open set is an open interval. Take for example $E:=(1,2)\cup(4,5)$. Then $E$ is open, as the union of open sets, but is not an interval.
