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I am reading Rudin's Real and complex analysis, he mentioned there exist situations in which $\bar{S}(x,r)$, does not contain all y with $\rho(x,y)\leq r$, $S(x,r)$ is the open ball of $x$. Could someone help to explain it?

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up vote 2 down vote accepted

Let $X = [-1, 1] \cup \{-2, 2\}$ with the Euclidean distance. $\overline{S}(0, 2)$ doesn't contain $\{-2, 2\}$.

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Or $\mathbb{N}$ with the Euclidean distance. – copper.hat Nov 13 '12 at 0:38

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