Definition:Perfect Set
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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
- Results about perfect sets can be found here.