Adherent Point and Closure of a Set¶

Definition¶

Let $(X,d)$ be a metric space and let $E$ be a subset of $X$ . A point $x_0$ in $X$ is termed an adherent point of $E$ if, for every $\varepsilon > 0$ $0$ , the ball $B(x_0, \varepsilon)$ intersects $E$ nonemptily. The set of all adherent points of $E$ constitutes the closure of $E$ , denoted as $\overline{E}$ or $\operatorname{cl}E$ .

Interpretation¶

An adherent point of a set is one that you can get arbitrarily close to from inside the set: every ball around it, however small, contains a point of the set. The closure is the original set together with all such points: everything you can touch or approach.

A shaded region E with three test points and small balls: one inside E, one on its edge whose ball still meets E, so it is adherent, and one outside whose ball misses E.

In the figure, the point inside $E$ is adherent because it belongs to $E$ itself; every point of $E$ is adherent. The point $x_0$ on the edge is adherent as well: each ball around it contains points outside $E$ but still meets $E$ . The grey point outside is not adherent, since one ball around it misses $E$ ; it is an exterior point.

Adherent points can be described with sequences: $x_0\in\overline{E}$ if and only if $x_0$ is the limit of a sequence in $E$ . If $x_0$ is adherent, pick $x_n\in B(x_0,\tfrac{1}{n})\cap E$ $E$ , so that $x_n\to x_0$ $x_{0}$ ; conversely, if $x_n\to x_0$ with all $x_n\in E$ $E$ , every ball around $x_0$ contains some $x_n$ .

The closure is the smallest closed set containing $E$ , and the points it adds are exactly the boundary points not already in $E$ . In Probability Theory, the support of a random variable with one of the usual densities is the closure of the region where the density is positive.

A common slip is to think that an adherent point must lie on the edge of $E$ or outside it. Every point of $E$ is adherent, including an isolated point such as $\boldsymbol{p}$ in the interactive example, so $E\subseteq\overline{E}$ always.

Key Results¶

We have $\overline{E} = \operatorname{int}E \cup \partial E = E \cup \partial E = X \setminus \operatorname{ext}E.$
A set is closed if and only if it contains all its adherent points.
The closure of a set $E$ is the intersection of all closed sets which contain $E$ , or equivalently, the smallest closed set that contains $E$ .

Examples¶

The closure of $(0,1)$ in $\mathbb{R}$ is $[0,1]$ . The endpoints $0$ and $1$ are adherent points even though they are not in the open interval.
If $E = \{1/n \colon n \in \mathbb{N}^+\}$ in $\mathbb{R}$ , the closure is $E \cup \{0\}$ ${0}$ . Zero is an adherent point approached by the sequence.

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 5

Aliases: closure point, contact point Keywords: closure of a set Symbols: element of, epsilon, intersection, natural numbers, partial, real numbers, subset, union

Last updated: 20 October 2026 03:02