# Definition:Perfect Set

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## 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.