Limit point

Definition

Let $X$ be a topological space, let $S \subseteq X$ and let $x \in X$. The point $x$ is a limit point of $S$ when each open set $U$ with $x \in U$ contains a point of $S$ other than $x$.

Remarks

The point $x$ need not be in $S$. In $\mathbb{R}$, the point $0$ is a limit point of $\{1/n : n \ge 1\}$. In a metric space, $x$ is a limit point of $S$ if and only if each ball $B(x, \varepsilon)$ contains a point of $S \setminus \{x\}$. The closure of $S$ is the union of $S$ and the set of its limit points. A point of $S$ that is not a limit point of $S$ is an isolated point of $S$.

Explanations

L1 human

A limit point is a point that other points of the set crowd around. For the set $\{1, 1/2, 1/3, \ldots\}$, the points crowd around $0$, so $0$ is a limit point, but $1$ is not.