Is $p$ is equivalent to the discrete metric ? Are they topologically equivalent?

Question

let $X$ be a nonempty set . Define a map on $X \times X$ by

$$p(x,y) = \begin{cases} 0,& \text{if }x=y \\ 2,& \text{if } x \neq y \end{cases}$$

Is $p$ is equivalent to the discrete metric? Are they topologically equivalent?

My attempt: I think $p$ will be equivalent to the discrete metric because both metric have same format define on non- empty set but I'm not sure about topologically equivalent

any hints/solution

thanks u

Answer 1

They are topologically equivalent because they have the same collection of open sets (namely all subsets of $\mathbb{R}$ are open in this metric, as in the discrete metric).