Equivalence of Definitions of Adherent Point

Theorem
The definitions of an adherent point are equivalent.

That is, let $\left({X, \tau}\right)$ be a topological space.

Let $A \subseteq X$.

Let $A^-$ denote the closure of $A$.

Then $x \in A^-$ iff, for every open neighborhood $U$ of $x$, the intersection $A \cap U$ is non-empty.

Proof
For a subset $S \subseteq X$, let $S^{\complement}$ denote the relative complement of $S$ in $X$.

The following equivalence holds:

The result follows from the Rule of Transposition.