# Limit point and interior point

Is any interior point also a limit point?

Judging from the definition, I believe every interior point is a limit point, but I'm not sure about it. If this is wrong, could you give me a counterexample?

(Since an interior point $p$ of a set $E$ has a neighborhood $N$ with radius $r$ such that $N$ is a subset of $E$. Obviously any neighborhood of $p$ with radius less than $r$ is a subset of $E$. Also, any neighborhood with radius greater than $r$ contains $N$ as a subset, so (I think) it is a limit point.)

• By the way, you tagged this "general-topology", but speak of radii of nhoods. Are you working in metric spaces? Aug 1 '12 at 22:06
• Yes. I just started analysis (Walter Rudin The principle of mathematical analysis) and I came up with this question when reading the chapter named the basic topology Aug 1 '12 at 22:09

No an interior point is not a limit point in general. Suppose you have $\mathbb{N}$ with the discrete metric. The point $1$ is a interior point since $\{1\}$ is an open set. However, $1$ is not a limit point of $\mathbb{N}$ because $\{1\}$ is an open neighborhood of $1$ which does not intersect $\mathbb{N}$ in any point beside $1$.
• A discrete topology is a space where every subset is open. Topologically equivalent, it is a space endowed with the discrete metric. Since every subset is open, the set $\{1\}$ is open. Aug 1 '12 at 22:18
• @Tengu in the language of Rudin, the set of points in $\mathbb{N}$ whose distance from 1 is less than, say, $\frac{1}{2}$, is exactly $\{1\}$, thus this single-element set is an open ball (and thus open). Aug 1 '12 at 22:23