let $d((a,b),(c,d))=d_{0}(a,c)+|b-d|$ be a metric on $\mathbb R^2$

let

X=$\{(x,0):x\in \mathbb R \}$ with the induced metric $d_{X}$

Y=$\{(0,y):y\in \mathbb R \}$ with the induced metric $d_{Y}$

A=$\{(x,y):x^2+y^2\leq4\}$

B=$\{(x,y):x^2+y^2<9\}$

i have to decide if the following subsets of X and Y are open,closed, neither or both.

1) $A \cap X$ with respect to $d_{X}$

2) $B \cap X$ with respect to $d_{X}$

3) $A \cap Y$ with respect to $d_{Y}$

4) $A \cap Y$ with respect to $d_{Y}$

can someone check i have this correct.

ive said that $d_{X}$, the restriction of d to X is just the discrete metric $d_{0}$ so i have both 1 and 2 are open and closed.

and $d_{Y}$ is just the euclidean metric so 3 is closed and 4 is open