Equivalence of Definitions of Adherent Point/Definition 1 iff Definition 3

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

Let $H \subseteq S$.

Necessary Condition
Let every open neighborhood $U$ of $x$ satisfy:
 * $H \cap U \ne \O$

Let $N$ be an arbitrary neighborhood of $x$.

By definition of a neighborhood:
 * $\exists V \in \tau: x \in V \subseteq N$

From Set is Open iff Neighborhood of all its Points, $V$ is an open neighborhood of $x$.

Thus:
 * $H \cap V \ne \O$

By the contrapositive statement of Subsets of Disjoint Sets are Disjoint:
 * $H \cap N \ne \O$

Because $N$ was arbitrary, it follows that every neighborhood $N$ of $x$ satisfies:
 * $H \cap N \ne \O$

Sufficient Condition
Let every neighborhood $N$ of $x$ satisfy:
 * $H \cap N \ne \O$

By definition, every open neighborhood $U$ of $x$ is a neighborhood of $x$.

So every open neighborhood $U$ of $x$ satisfies:
 * $H \cap U \ne \O$