Undergrad Open balls dense in closed balls in Euclidean space

[psie]

TL;DR

I am working a problem in Gamelin and Greene's book on topology. They ask about whether closed balls are closed sets (which they are), but moreover if the closure of an open ball is a closed ball. They make a statement concerning this which I don't understand.

Any set with at least two elements and equipped with the discrete metric is a counterexample to the claim that the closure of an open ball is a closed ball. Yet, in the back of the back book where they present solutions to some of their exercises, they write:

The statement about open balls being dense in closed balls holds in $\mathbb R^n$, but it does not hold in metric spaces in general.

I feel silly for asking, but I can not make sense logically of the first part of the sentence. First they say it holds in $\mathbb R^n$, but not in metric spaces in general. What could they mean by it holds in $\mathbb R^n$? My understanding is that the statement holds in $\mathbb R^n$ equipped with a metric derived from a norm, but not otherwise. Is this correct?

 

[pasmith]

\mathbb{R}^n, without more, means \mathbb{R}^n with the euclidean norm.

 

[WWGD]

Somewhat related: The standard metric in $\mathbb R^n$ is derived from an inner-product$<,>$, which gives rise to a norm $||.||$, through , $||v||=<v,v>^{1/2}$, and a metric $m(x,y):=<x-y, x-y>^{1/2}$