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.

Necessary Condition
We use the Rule of Transposition.

Let $x \in X \setminus A^-$.

Since $A^-$ is closed, it follows that $X \setminus A^-$ is an open neighborhood of $x$.

By the definition of the closure, we have $A \subseteq A^-$.

Hence, $A \cap \left({X \setminus A^-}\right)$ is empty.

Sufficient Condition
We use the Rule of Transposition.

Let $x \in X$, and let $U$ be an open neighborhood of $x$ such that the intersection $A \cap U$ is empty.

Then $A \subseteq X \setminus U$.

Since $X \setminus U$ is closed, it follows by Set Closure is Smallest Closed Set that $A^- \subseteq X \setminus U$.

Since $x \in U$, it follows by the Rule of Transposition that $x \notin A^-$.