Equivalence of Definitions of Limit Point

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.

$({1}) \iff ({2})$
The closure of $A$ is defined as the union of the set of all isolated points of $A$ and the set of all limit points of $A$.

The rest then follows directly from Equivalence of Definitions of Isolated Point.

$({2}) \iff ({3})$
Follows directly from Equivalence of Definitions of Adherent Point.

Also see

 * Definition:Limit Point of Set