Does the boundary points of a set depend on the metric used?

Question

Suppose we have a closed plane curve in $\mathbb{R^2}$, now in the usual $2$ norm, we say the boundary of the points contain inside this bounding curve is the interior and the plane curve itself is the boundary. But, suppose we change the metric (say eg another p norm), would the same interior (as set) have a different boundary (different from the plane curve)?

I think maybe it could because boundary and interior points are distinguished by balls containing them and ball are dependent on the metric..

Answer 1

Any two metrics on an Euclidean space induced by a norm induce the same topology. So open sets, closed sets, closure, boundary etc do not change. But if the metric is not induced by a norm the the boundary can change. (Look at the discrete metric. The boundary of any set is empty in this case!)