Closed Set

In Topology

In a topological space $(X,T)$, a subset $A\subseteq X$ is closed if its complement $X-A$ is open.

Equivalently, the collection of closed sets in $X$ satisfies:

• $\varnothing$ and $X$ are closed
• Arbitrary intersections of closed sets are closed: if $\{C_i{\}}_{i\in I}$ is a collection of closed sets, then ${\bigcap }_{i\in I}{C}_i$ is closed
• Finite unions of closed sets are closed: if $C_1,\dots ,C_n$ are closed, then $C_1\cup \cdots \cup C_n$ is closed

These properties are dual to the axioms for open sets, obtained by interchanging unions with intersections and arbitrary with finite.

In Metric Space

In a metric space $(X,d)$, a set $A\subseteq X$ is closed if it contains all its limit points.

Equivalently, $A$ is closed if for every convergent sequence $(x_n)$ in $A$ (with $x_n\in A$ for all $n$), the limit of the sequence is also in $A$: $\text{if }{x}_n\in A\text{ for all }n\text{ and }{x}_n\to x,\text{ then }x\in A$

Relationship Between Definitions

Every metric space $(X,d)$ induces a topology where the open sets are those satisfying the metric space definition of openness. Under this topology:

• A set is closed in the topological sense (complement is open) if and only if it is closed in the metric sense (contains all limit points)
• The closure $\overline{A}$ is the smallest closed set containing $A$
• A set $A$ is closed if and only if $A=\overline{A}$

Examples

• In $\mathbb{R}$ with the standard metric, $[0,1]$ is closed, $(0,1)$ is not closed
• In any topological space, $\varnothing$ and $X$ are both open and closed
• The set $\mathbb{Q}$ of rationals is neither open nor closed in $\mathbb{R}$
• In a discrete space, every subset is both open and closed
• The set $\{1/n\mid n\in {\mathbb{N}}^{+}\}\cup \{0\}$ is closed in $\mathbb{R}$ because it contains its only limit point $0$

Properties

• A set can be both open and closed (called clopen)
• A set can be neither open nor closed
• The complement of a closed set is open, and vice versa
• Closed sets are characterized by containing their boundary

Related Concepts

• Open Set (Topology)
• Topological Space
• Metric Space
• Closure (Metric Space)
• Limit Point
• Boundary (Topology)
• Interior (Topology)