Limit point intuition Quoting Rudin,
"A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q\not=p : q \in E$."
This would imply that the points in an open ball would all be limit points, since for any $p$ in $E$ there are $q$ such that $d(p,q) < r$ for all $q \in E$. So E is also a neighborhood of the open ball.
Is my intuition correct? What can be improved about this statement?

The following is likely to be not relevant, since the space being discussed is undoubtedly $\mathbb{R}^n$ for some $n$. But it is too long for a comment.
Suppose that our space $S$ consists of the points $0$, and $\frac{1}{k}$, where $k$ ranges over the positive integers. Use the ordinary distance function $|x-y|$.
Then $0$ is a limit point of $S$, since every interval about $0$ contains a point of $S$ different from $0$. However, if we pick any $\frac{1}{k}$ in this interval, then there is an interval about $\frac{1}{k}$ that contains no point of $S$ other than $\frac{1}{k}$. 
