are two metrics with same compact sets topologically equivalent?

are two metrics with same compact sets topologically equivalent ?

I think if the cardinal of set is finite then we have one metric that is the discrete metric and every metric on this set is equivalent with the discrete metric.

now let $X$ be infinite set , in this case I consider $X= \Bbb{N}$ and $d(x,y)=|x-y|$ and k(x,y) is discrete metric ,k and d have same compact sets and are topologically equivalent because any single set is open set, are two metrics with same compact sets topologically equivalent ? (on infinite set)

Yes, because in particular they have the same convergent sequences (a convergent sequence with its limit is a compact subset). And so the same closed sets (for metric spaces $X$, a subset $C$ is closed iff for every sequence from $C$ that converges in $X$ has its limit in $C$), and so the same open sets as well.