Is an isolated point in $\mathbb{R}^d$ a limit point? I just read the definition of limit point and isolated point in the book Real Analysis: measure theory, integration and hilbert spaces by e.m. stein and r. shakarchi
A point $x\in\mathbb{R}^d$ is a limit point of $E$ if for every $r>0$, the ball $B_{r}(x)$ contains points of $E$. ( Does it mean $B_{r}(x)\cap E \neq \emptyset ?$ )
 An isolated point of $E$ is a point $x\in E$ s.t. there exists an $r>0$ where $B_{r}(x)\cap E$ is equal to $\{x\}$ 
Since the definition of limit point here does not require $B_{r}(x)$ contains points of $E\backslash$$\{x\}$, then an isolated point of $E$ is a limit point of $E$ since for every $r>0$, the ball $B_{r}(x)$ contains a point $x\in E$. I was confused about that. (Or the "points" in the definition of limit point means at least two different points?)

The way these definitions are written here, yes, it looks like isolated points become limit points.
This is, however, not the conventional definition of limit points. We usually require that $B_r(x)$ contains points of $E$ in addition to $x$ itself. With the conventional definition, isolated points are not limit points. In fact, the isolated points of $E$ are exactly the points of $E$ that are not limit points (note that also points outside $E$ can be limit points of $E$).
