Definition:Closed Set/Metric Space

Definition

Let $M = \left({A, d}\right)$ be a metric space.

Let $H \subseteq A$.

Definition 1

$H$ is closed (in $M$) if and only if its complement $A \setminus H$ is open in $M$.

Definition 2

$H$ is closed (in $M$) if and only if every limit point of $H$ is also a point of $H$.

Also see

• Equivalence of Definitions of Closed Set in Metric Space

• Definition:Closed Set (Complex Analysis)
• Definition:Closed Set (Real Analysis)