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. 1.

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

2. 2.

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

3. 3.

Every nonempty family of closed subsets has a minimal element.

4. 4.

Every nonempty family of open subsets has a maximal element.

5. 5.

Every subset of $X$ is compact.

Title alternative characterizations of Noetherian topological spaces AlternativeCharacterizationsOfNoetherianTopologicalSpaces 2013-03-22 14:16:25 2013-03-22 14:16:25 yark (2760) yark (2760) 10 yark (2760) Theorem msc 14A10