Limit point (accumulation point, cluster point)

Let $(X,d)$ be a metric space and let $A\subseteq X$. A point $x\in X$ is a limit point (also called an accumulation point or cluster point) of $A$ if

(see open ball ).

Limit points are the points that can be approached by elements of $A$ distinct from the point itself. They determine closedness (a set is closed iff it contains all its limit points) and appear throughout analysis. Compare with isolated points .

Examples:

• In $\mathbb{R}$, every point of $[0,1]$ is a limit point of $(0,1)$; in particular, $0$ and $1$ are limit points of $(0,1)$.
• In $\mathbb{R}$, $0$ is a limit point of $\{1/n:n\in\mathbb{N}\}$.
• In $\mathbb{R}$, the set $\mathbb{Z}$ has no finite limit points (each integer is isolated).