Equivalence of Definitions of Noetherian Topological Space

Definition 2 implies Definition 4
Let $A \subseteq \tau$ be non-empty. $A$ has no maximal elements.

Let $X_1 \in A$.

As $X_1$ is not a maximal element, there exists a $X_2 \in A$ such that:
 * $X_1 \subsetneq X_2$

Then, as $X_2$ is again not a maximal element, there exists a $X_3 \in A$ such that:
 * $X_2 \subsetneq X_3$

Repeating this procedure, we obtain:
 * $X_1 \subsetneq X_2 \subsetneq X_3 \subsetneq \cdots$

This contradicts the ascending chain condition.

Therefore $A$ has a maximal element.