Open sets Are balls? Is any open set a ball? 
So instead of open sets we can talk about balls or disks?
Is that correct? Because ihave seen definitions with open sets Like the definition of a surface.So is it sufficient instead of talking about  generally open sets simplify things thinking it as b
So the question is: given a subset $U \subseteq \mathbb{R}^n$, we say that a subset $K$ of $U$ is open in $U$ if there exists an open set $M$ of $\mathbb{R}^n$  such that $K = M \cap U$ .  In the definition, can we change "open set $M$ of $\mathbb{R}^n$" to "open ball $M$ in $\mathbb{R}^n$?"

There are many definitions of open sets. If you have a topology, then all elements of it are defined to be open, and they do not need to be balls.
If you have a metric, then a set $A$ is open if for all $x\in A$, there exists an open ball $B(a,r)$ (an open ball is defined as $B(a,r)=\{x: d(x, a)<r\}$) such that $x\in B(a,r)\subseteq A$.
This means that in a metric space, all open balls are also open sets, but not all open sets are open balls. For example, $(0,1)\cup (3,4)$ is not an open ball, but it is an open set.
