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 $\mathbb{R}^n$?"

No. Draw some simple figures in the plane as such: 
But by the definition of an open set, to each point $x \in U$, there exists an open ball around that point completely contained in the set.
Thus,
$$U = \bigcup_{x \in U} B_{\epsilon(x)}(x)$$
where $\epsilon(x)$ is a radius that keeps the open ball around $x$ contained in $U$.
Edit: Your edit is a question about the relative topology. Take an open subset $K$ of $U$ above that doesn't look like a ball and notice that no ball $M$ exists such that $K = U \cap M$. Do some reading on the concept of the relative topology. Thus, no it does not suffice.
