Closed set

Let $(X,d)$ be a metric space . A subset $F\subseteq X$ is closed if its complement $X\setminus F$ is open .

In metric spaces, closedness is equivalent to several other important properties (e.g., containing limits of convergent sequences from the set). Closed sets are stable under intersections and are used in defining closures and compactness .

Examples:

• In $\mathbb{R}$, the interval $[a,b]$ is closed.
• In $\mathbb{R}$, $\mathbb{Z}$ is closed (its complement is a union of open intervals).
• In any metric space, $\varnothing$ and $X$ are closed.