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.