Definition:Perfect Set

From ProofWiki
Jump to: navigation, search

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 $T = \left({S, \tau}\right)$ is a subset $H \subseteq S$ such that:

$H$ is dense-in-itself.
$H$ contains all its limit points.


Also see

  • Results about perfect sets can be found here.