Equivalence of Definitions of Adherent Point

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

Let $A \subseteq S$.

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

It is required to be shown that $x \in A^-$, for every open neighborhood $U$ of $x$, the intersection $A \cap U$ is non-empty.

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

We have that:

Thus:
 * $x \in U \iff x \notin A^-$

The result follows from the Rule of Transposition.