Equivalence of Limit Point Definitions

Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq T$.

The following definitions for a limit point of $H$ in $T$ are equivalent:

$(1): \quad x \in S$ is called a limit point of $H$ if every open set $U \in \vartheta$ such that $x \in U$ contains some point of $H$ other than $x$.

$(2): \quad x \in S$ is called a limit point of $H$ if $x$ belongs to the closure of $H$ but is not an isolated point of $H$.

$(3): \quad x \in S$ is called a limit point of $H$ if $x$ is an adherent point of $H$ but is not an isolated point of $H$.

$(4): \quad x \in S$ is called a limit point of $H$ if there is a sequence $\left\langle{x_n}\right\rangle$ such that $x$ is a limit point of $\left\langle{x_n}\right\rangle$.