Definition:Closed Set/Topology/Definition 2

Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$.

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

That is, by the definition of the derived set:

$H$ is closed (in $T$) if and only if $H' \subseteq H$

where $H'$ denotes the derived set of $H$.

Also see

• Equivalence of Definitions of Closed Set

• Results about closed sets can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): closed set (of points)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): closed set (of points)