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.
$S$ is not an interval, open, or otherwise. 

An interval $I$ in any ordered set has the property:


  If $a&lt;b&lt;c$ 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.
Difference-1 if we cosider thee sequence with element of  the subset of a open interval then the sequence is monotone increasing ......but if we consider a sequene with the element of the subset of an open set then the sequence is either monotone increasing or monotone decreasing .....