Q:

What is difference between open set and open interval?

Let $\tau$ be the Euclidean topology defined on $\mathbb R$. If we define a set $S = (2,3) \cup (5,6)$. Then is the set $S$ an open set and open interval on $\tau$?
As per definition of open set,

A subset $A$ of $\tau$ is open set if  $\forall x \in A$, $\hspace{5pt} \exists  \hspace{3pt} a,b $ such that $x \in (a,b) \subseteq A$ 

As per this definition, we can find $(a,b) \in S$ such that $x \in (a,b)$. $x$ is any number in set $S$. So, set $S$ is open set. 
I am not sure whether I correctly proved why set $S$ is open set. But I do not know how to prove set $S$ is open interval. 

A:

$S$ is not an interval, open, or otherwise. 
An interval $I$ in any ordered set has the property:

If $a<b<c$ and $a,c\in I$ then $b\in I$.

Your set $S$ is therefore open, but not an interval. It is the union of two open intervals.

