alternative characterizations of Noetherian topological spaces
Let $X$ be a topological space^{}. The following conditions are equivalent^{} conditions for $X$ to be a Noetherian topological space:

1.
$X$ satisfies the descending chain condition^{} (http://planetmath.org/DescendingChainCondition) for closed subsets.

2.
$X$ satisfies the ascending chain condition^{} (http://planetmath.org/AscendingChainCondition) for open subsets.

3.
Every nonempty family of closed subsets has a minimal element.

4.
Every nonempty family of open subsets has a maximal element.

5.
Every subset of $X$ is compact^{}.
Title  alternative characterizations of Noetherian topological spaces 

Canonical name  AlternativeCharacterizationsOfNoetherianTopologicalSpaces 
Date of creation  20130322 14:16:25 
Last modified on  20130322 14:16:25 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  10 
Author  yark (2760) 
Entry type  Theorem 
Classification  msc 14A10 