Equivalence of Definitions of Noetherian Topological Space

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

That is:
 * $\forall X \in A : \set {Y \in A : X \subsetneq Y} \ne \O$

Thus by axiom of choice there is a mapping:
 * $f : A \to A$

such that:
 * $\forall X \in A : \map f X \subsetneq X$

Choose $X_1 \in A$ arbitrarily.

Recursively, for $i = 1,2,\ldots$ let:
 * $X_{i + 1} := \map f {X_i}$

so that:
 * $X_1 \subsetneq X_2 \subsetneq X_3 \subsetneq \cdots$

This contradicts the ascending chain condition.

Therefore $A$ has a maximal element.