Open and Closed Sets in a Metric Space¶

Definition¶

For a metric space $(X,d)$ and its subset $E$ :

$E$ is closed if it includes all of its boundary points, i.e., $\partial E \subseteq E$ $E$ .
$E$ is open if it doesn't include any of its boundary points, i.e., $\partial E \cap E = \varnothing$ $\emptyset$ .
$E$ is neither open nor closed if it includes some but not all of its boundary points.

Interpretation¶

Openness and closedness are decided by the boundary of a set. Open sets keep their boundary points out, closed sets keep them in, and sets that include some boundary points but not others are neither open nor closed.

Three discs: an open disc with dashed rim, a closed disc with solid rim, and a disc whose rim is partly solid and partly dashed, which is neither open nor closed.

All three discs in the figure have the same boundary, the circle. A dashed rim means that the circle is excluded, as for the open disc $\{d(\boldsymbol{x},\boldsymbol{x}_0)<r\}$ $r}$ ; a solid rim means it is included, as for the closed disc $\{d(\boldsymbol{x},\boldsymbol{x}_0)\le r\}$ $r}$ . The third disc contains half of its rim, so it fails both conditions.

The Key Results give two working descriptions. A set is open if and only if every point of it is an interior point: around each point there is a ball inside the set, so one cannot reach the outside by a small step. A set is closed if and only if it contains the limits of its convergent sequences, so one cannot leave it by passing to a limit. Since $E$ is open exactly when its complement is closed, every statement about open sets has a mirror statement about closed sets.

A common slip is to treat "open" and "closed" as opposites. The interval $[0,1)$ in $\mathbb{R}$ is neither, while $\varnothing$ and $X$ are always both. Showing that a set is not open therefore does not show that it is closed.

Open sets are also the raw material of a topology, which keeps the open sets and forgets the distances; later, continuity and compactness are expressed through open sets as well.

Key Results¶

A set $E$ is open if and only if $E = \operatorname{int}E$ $int E$ .
A set $F$ is closed if and only if every convergent sequence $(x_i)_{i=1}^{\infty}$ within $F$ converges to a limiting point lying in $F$ .
A set $E$ is open if and only if its complement $E^\complement=X\setminus E$ is closed.

Examples¶

The open metric ball $B(x_0, \varepsilon)$ is open.
$\varnothing$ is both open and closed. It has no boundary points to include or exclude.
Under the discrete metric, every set is both open and closed. Each singleton set has a ball containing only itself.

Prerequisites¶

Boundary of a Set

Sources¶

Math III: Analysis · Lecture 5

Aliases: open set, closed set, open subset, closed subset Symbols: epsilon, infinity, intersection, partial, real numbers, subset

Last updated: 20 October 2026 03:02