According to Rudin's Principles of Mathematical Analysis:
A point $p$ is a limit point of the set $E$ if every neighbourhood of $p$ contains a point $q\ne p$ such that $q\in E$
If $p\in E$ and $p$ is not a limit point of $E$, then $p$ is called an isolated point of $E$
A point $p$ is an interior point of $E$ if there is a neighbourhood $N$ of $p$ such that $N\subset E$
Consider $R^3$ with the usual distance metric. Off course, a limit point might not be contained in $E$. But, consider the case where $p\in E$. This means that:
- all neighbourhoods of $p$ will contain some $q$ that is also in $E$
- in fact there will be a neighbourhood of $p$ that is a subset of $E$
So, $p$ it is both a limit point and an interior point of $E$. In consequence, $E$ has no isolated points?
Could you give me some examples of spaces where one can identify points that are only limit, isolated and interior.