I'm having some difficulty understanding what exactly "open ball" means when it comes to dealing with subsets of $\mathbb R^2$, and likely by extension any metric space.
Consider the subsets of $\mathbb R^2$, $$A = \{(x,y) : x+y=1\}$$ $$B = \{(x,y) : x,y\in\mathbb Q\}.$$
Let $\langle M,\rho\rangle$ be a metric space. If $a\in M$ and $r>0$, then the open ball of radius $r$ about $a$, $B(a;r)$, is defined as $$B(a;r) = \{x\in M : \rho(x,a) < r\}.$$
I'm wondering how to determine whether or not these sets are open/closed. We say that a subset $G$ of $M$ is open if for every $x\in G$, there exists a number $r>0$ such that the entire open ball $B(x;r)$ is contained in $G$.
So when considering a subset of a metric space, can an open ball of $G$ contain points of $M$ that are outside of $G$? For example, consider the picture:
This is an image of a line such as the one in $A$. Is the open ball the yellow and purple space, or is it just the purple part?
The definition is just a little confusing to me.