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Let $X$ be a metric space, and $A\subset X$. I read that if $x\in X\setminus A$ is the limit point of $\{x_i\}$ in $A$, then it is not necessarily true that $x$ is a limit point of $A$.

How is this possible? If $x$ is the limit point of $\{x_i\}$, then every open set containing $x$ contains points from $\{x_i\}$, and hence points from $A$.

Thanks in advance!

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Where did you read this? It is incorrect. You're quite right.

Cameron Buie
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  • A book on Metric Spaces by Babu Ram. Probably a misprint or something. –  Aug 10 '13 at 19:39
  • I would think so. If the word "not" were removed, it would be just fine. – Cameron Buie Aug 10 '13 at 19:43
  • @CameronBuie Maybe the author means that the constant sequence ${a,a,a,a,a,\ldots}$ has $a$ as a limit point, but $a$ might be isolated in $A$. – Pedro Aug 10 '13 at 20:04
  • @Peter: That was my initial thought, but it was specified that $x$ was not a point of $A$, while the sequence consists of points of $A$ and converges to $x$. – Cameron Buie Aug 10 '13 at 20:06
  • Ah! I think I see what you mean, now. @Ayush: Peter seems to be pointing out that if $x\in A$ and there is a sequence of points of $A$ converging to $x$, then it needn't follow that $x$ is a limit point of $A$. Consider a discrete metric space, for example. – Cameron Buie Aug 10 '13 at 20:33