Equivalence of Definitions of Noetherian Topological Space

Theorem

The following definitions of the concept of Noetherian Topological Space are equivalent:

Definition 1

A topological space $T = \struct {S, \tau}$ is Noetherian if and only if its set of closed sets, ordered by the subset relation, satisfies the descending chain condition.

Definition 2

A topological space $T = \struct {S, \tau}$ is Noetherian if and only if its set of open sets, ordered by the subset relation, satisfies the ascending chain condition.

Definition 3

A topological space $T = \struct {S, \tau}$ is Noetherian if and only if each non-empty set of closed sets has a minimal element with respect to the subset relation.

Definition 4

A topological space $T = \struct {S, \tau}$ is Noetherian if and only if each non-empty set of open sets has a maximal element with respect to the subset relation.

Proof

Definition 2 implies Definition 4

Let $A \subseteq \tau$ be non-empty.

Aiming for a contradiction, suppose $A$ 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.

$\Box$

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