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To learn the definition of a closed set - a set which contains all its limit points, I want to fully understand the concept of a limit point. What exactly is a limit point?

I think that a limit point is a point 'at the end' of (each subset) of the set. For example $a$ and $b$ if the set is $[a,b]$, or the points $x^2+y^2 = 1$ for the set which is the (filled) unit circle.

In my book, the definition is as follows:

A point $x$ is a limit point of a set $A$ is every $\epsilon$-neighborhood $V_{\epsilon}(x)$ of $x$ intersects the set $A$ in some point other then $x$

What is so special about a limit point? The only thing I read is that if $x$ is a limit point of $A$, there must be another $y \in A$, such that if $\epsilon\gt0$, then $|x-y|<\epsilon$

But why is that not true for 'non'-limit points? How can I distinguish these points from point that are not limit points in the set ?

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marked as duplicate by BenjaLim, Hagen von Eitzen, Micah, Henry T. Horton, Brandon Carter Feb 7 '13 at 17:04

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im sorry. did not see that –  MSKfdaswplwq Feb 7 '13 at 16:38
    
Be careful, for every $\epsilon >0$ there is another $y\in A$ such that $|x-y|<\epsilon$ is not the same as: there is another $y\in A$ such that if $\epsilon> 0$, then $|x-y|<\epsilon$. Note in particular, that the second one leads to $x=y$ by letting $\epsilon$ tend to $0$, hence it is a false statement. –  1015 Feb 7 '13 at 16:41
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Think of it this way (at least for metric spaces): a limit point is a point that has points around it of arbitrary closeness, so you can make a sequence of distinct points that converges to the limit point.

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