Find the boundary and interior for each of the following subsets of $R^2$
$A= \{(x,y): y=0 \}$ in $\mathbb{R}^2$
I got the answer 
as I'm not getting the red circle as I have not getting. What does it indicate, as I have taken it from http://math.ucr.edu/~res/inprogress/sol20828.pdf
It could mean the open ball with respect to the Euclidean norm centered at $(x,y)$ with radius $|y|> 0$, more commonly denoted by $B((x,y), |y|)$.
Indeed we have $B((x,y), |y|) \subseteq \mathbb{R}^2 - A$ since
$$(x', y') \in B((x,y), |y|) \implies |y| - |y'|< |y'-y| \le \|(x', y') - (x,y)\|_2 < |y|$$
and therefore $|y'| > 0$ so $y' \ne 0$.