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Say $x$ is a limit point of a set $A$.

That means that for any $r > 0$, $B_r(x)$ contains a point of $A$ other than $x$ itself.

Or I often see it stated as $B_r(x) \cap A \neq \varnothing$ and these statements are taken to be equivalent.

But they don't seem equivalent to me. Say I have a set $A = \{2\} \cup [4,6]$ in $\mathbb{R}$.

So if we consider the point $2$, the first statement above says that $2$ will not be a limit point. However if we take the $B_r(x) \cap A \neq \varnothing$, $2$ will be a limit point because that open ball contains $2$ itself.

The $B_r(x) \cap A \neq \varnothing$ definition seems invalid as it doesn't take account of the fact that we only want to consider points in the ball other than $x$. So have I missed something or is $B_r(x) \cap A \neq \varnothing$ not a valid statement when it comes to defining a limit point of a set?

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2 Answers

The second one should be stated as $B^*_r(x)\cap A\neq \emptyset$, where $B_r^*(x)$ is the deleted neighborhood of $x$ - $B_r^*(x)=B_r(x)\backslash\{x\}$. Then they are equivalent. Otherwise, as your example illustrates, they are not.

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Some authors differ between a limit point and an isolated point, some don't. That's the difference.

In other words for some authors limit points (of an set $A\subset X$) are those points where you can find $(x_n)_n\subset A$ and $\lim\limits_{n\rightarrow\infty}x_n = x\in X$ (what would leed to your 2nd definition) and some author authors require in addition to that, that $\lim\limits_{n\rightarrow\infty, x_n\neq x}x_n = x\in X$ (what would leed to your 1st definition).

Both definitions are ok (but not equivalent as you have already shown) and it depends on the context (and the authors preferences) which you should prefer. For example with respect to the first definition you can say:

  • A set $A\subset X$ is closed $\Longleftrightarrow$ $\left(x\right.$ is a limit point of $A$ $\Longrightarrow$ $\left.x\in A\right)$

with respect to the second definition you could even say something like

  • A set $A\subset X$ is closed $\Longleftrightarrow$ $\left(x\right.$ is a limit point of $A$ $\Longleftrightarrow$ $\left.x\in A\right)$
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