Definition:Perfect Set

This page is about Perfect in the context of Topology. For other uses, see Perfect.

Definition

Definition 1

A perfect set of a topological space $T = \left({S, \tau}\right)$ is a subset $H \subseteq S$ such that:

$H = H'$

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

That is, where:

every point of $H$ is a limit point of $H$

and

every limit point of $H$ is a point of $H$.

Definition 2

A perfect set of a topological space $T = \left({S, \tau}\right)$ is a subset $H \subseteq S$ such that:

$H$ is a closed set of $T$

$H$ has no isolated points.

Definition 3

A perfect set of a topological space $T = \struct {S, \tau}$ is a subset $H \subseteq S$ such that:

$H$ is dense-in-itself.

$H$ contains all its limit points.

Also see

• Equivalence of Definitions of Perfect Set

• Results about perfect sets can be found here.