Metric spaces/interior, boundary, closure

Question

Define $X=[-1,1]×${$0$}$ \ ⊂ \mathbb{R^2}$. How to determine the interior, closure and boundary of this subset with respect to the euclidean distance and the discrete metric on $\mathbb{R^2}$?

For the euclidean distance I got:

$\mathring{A}=(-1,1)×${$0$}, since $B_\varepsilon(x,y)⊂(-1,1)$ and

$\bar{A}=[-1,1]×${$0$}

$\partial{A}=${$-1,1$}

I'm not sure if this works. For the discrete metric I don't know how to find these sets.

Answer 1

You are wrong. The set $A$ contains no open ball, and therefore $\mathring A=\emptyset$. On the other hand, $A$ is closed. Therefore, $\overline A=A$ and $\partial A=A$.

With respect to the discrete metric, every set is open and every set is closed. So, $\overline A=\mathring A=A$ and so $\partial A=\emptyset$.