I Open balls dense in closed balls in Euclidean space

psie
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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?
 
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\mathbb{R}^n, without more, means \mathbb{R}^n with the euclidean norm.
 
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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}##
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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