Equivalence of Definitions of Limit Point/Definition (1) iff Definition (4)

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

Let $A \subseteq S$.

Proof 1
The following equivalence holds:

The result follows from the Rule of Transposition.

Proof 2
The following equivalence holds:

There exists an open neighborhood $U$ of $x$ such that $A \cap \paren {U \setminus \set x} = \O$

By, $\map \complement A \cup \set x$ is a neighborhood of $x$

The result follows from the Rule of Transposition.