Why is the closure and interior of this set what they are under this particular metric? I'm revising for an upcoming exam and I'm still struggling to understand metrics. This revision question gives the answer with some explanation, but I still don't understand why it is what it is.
Question

Write the closure and interior of $$A = \left\{(x,y)∊\mathbb{R^2} : x\geq 2 \text{ and } y \leq 0 \right\}$$ for mixed metric, $d$ given by $$d((x_1,x_2),(y_1, y_2)) = d_0(x_1 , y_1 ) + d^{(1)} ( x_2 , y_2 ) $$ $d_0$ is the discrete metric and $d^{(1)}$ is the Euclidean metric in $1$ dimension.

Thanks
Answer

Now consider the mixed metric d. In the $x$-direction we see that $d$ behaves like the discrete metric and in the $y$-direction $d$ behaves like the Euclidean metric. We also know that any set is both $d_0$-open and $d_0$-closed.
Therefore we have:
$$Int_d(A) = \left\{( x , y ) ∊ \mathbb{R^2} : x\geq 2 \text{ and } y < 0 \right\} $$
and
$$Cl_d ( A ) = \left\{ ( x , y ) ∊ \mathbb{R^2} : x \geq 2 \text{ and } y \leq 0 \right\}$$

Why do the different directions behave like a certain metric?  How can I tell if it is not immediately obvious from the reason why?  Why is $x \geq 2$ in the interior rather than $x > 2?$  I'm guessing it is related to the set being both $d_0$-open and $d_0$-closed?
My ideal answer would be going through the question step by step with as much justification for each step as you are willing to give me.  With that said, I'll take whatever anyone is willing to offer.  Hopefully, having it explained to me step by step in an example would make it click, because I struggle when working with metrics other than the Euclidean ones.
 A: I think that if you figure out what that metric means, the question will solve itself. Let's try to explicitly calculate that metric.
The discrete metric is given by:
$$d_0(x_1,x_2) =
\begin{cases}
1, &\text{if $x_1 \ne x_2$}\\
0, &\text{if $x_1 = x_2$}\\
\end{cases}$$
and the 1-dimensional Euclidean metric given by:
$$d^{(1)}(y_1,y_2)=|y_2-y_1|$$
Thus your metric is given by:
$$d((x_1,y_1),(x_2,y_2))=
\begin{cases}
1+|y_2-y_1|, &\text{if $x_1 \ne x_2$}\\
|y_2-y_1|, &\text{if $x_1 = x_2$}\\
\end{cases}$$
Now, take two points with $(a,y_1)$ and $(a,y_2)$. Thus $d((a,y_1),(a,y_2))=|y_2-y_1|$, which is the Euclidean metric in the $y$-direction.
Taking two points $(x_1,b)$ and $(x_2,b)$, $x_1\ne x_2$, we get $d((x_1,b),(x_2,b))=1-|b-b|=1$. Of course, if we take $x_1=x_2$, then the distance is zero. Hence, it is the discrete metric in the $x$ direction.
A: If $(x_1,x_2) \in A$ with $x_2 \neq 0$, then $B((x_1,x_2),\min(1,|x_2|)$ stays inside $A$, and so it's an interior point of $A$. If you know the product topology, it's even easier to see the closure and interior. 
