# Confusing definition of limit points

I found this definition on wikipedia.

A point x in X is a limit point of S if every neighbourhood of x contains at least one point of S different from x itself.

But doesn't this just mean it could be pretty much any point? Not necessarily anywhere near a boundary/limit of S? Just a random point in S.

For example pick the point, 2, in S (the standard topology of R between (1,3)). Every neighbourhood of x will contain a point in S that is not x.

What am I missing, this notion of a limit point seems pointless..?

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EVERY nbhd of x. – user254665 Mar 20 at 12:22

Intuitively, when we say that some $x \in X$ is a limit point of $S$, we mean that that there are infinitely many elements of $S$ that are densely packed near $x$. Indeed, we can get arbitrarily close to $x$ using only elements of $S$.

For example, you may have heard that $\mathbb Q$ is dense in $\mathbb R$. A consequence of this fact is that any irrational number, such as $x = \pi$, is a limit point of $S = \mathbb Q$ and thus can be approximated to an arbitrary precision using only rational numbers such as: $$3, 3.1, 3.14, 3.141, 3.1415, 3.14159, \ldots$$

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In your example the set $(1,3)$ is open, so every point is a limit point - try to proof this for a arbitrary open set for better understanding. But now, for example, have a look at the set $(0,1] \cup \{2\} \cup [4,8)$. Here is not every point a limit point - try to find them. So this notion is not pointless only because there are some sets in which every point is a limit point.

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haha yea, now you mention it, it does seem a little silly to say "limit points are pointless only because there are some sets in which every point is a limit point." – Alexander Telfar Mar 20 at 8:17

The word limit in the definition of limit point its not related to the concept of topological boundary. It is related to the fact that it is the limit of a sequence of points in $S$ different from $x$.

Also, not every point of $S$ is a limit point. Isolated points like $2$ in $(0,1)\cup\{2\}$ are not limit points.

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cool thanks. I think I get it now, I was just considering continuous open sets for which limit point don't help much, but there are plenty of other uses in discontinuous sets. – Alexander Telfar Mar 20 at 8:18

Everyone has already written nice answers. I just wanted to show some pictures to build an idea bout the definition . If you see for $(0,1)$ take any neighbourhood of $0$ (as it says every nbhd of $x$ in the definition of the limit point) you will always find a point of $(0,1)$

For the second one if you see you can find a sufficiently large $\epsilon$ such that the intersection between $[0,1]$ and $B_{\epsilon}(2)$ will be nonempty but not for every neighbourhood.

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