Open balls dense in closed balls in Euclidean space

In summary, the concept of open balls being dense in closed balls in Euclidean space asserts that for any closed ball and any point within it, there exists an open ball centered at that point which is entirely contained within the closed ball. This property highlights the topological relationship between open and closed sets in Euclidean spaces, demonstrating that open balls can approximate closed balls arbitrarily closely. The result is significant in analysis and topology, as it illustrates the nature of convergence and limits within these spaces.
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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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[itex]\mathbb{R}^n[/itex], without more, means [itex]\mathbb{R}^n[/itex] 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}##
 
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