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.

单词	AlternativeCharacterizationsOfNoetherianTopologicalSpaces
释义	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.
随便看	NormalityOfSubgroupsOfPrimeIndex normality of subgroups of prime index Normalizer normalizer NormalizerCondition normalizer condition NormalizingReduction normalizing reduction NormalLine normal line NormalMatrix normal matrix NormalModalLogic normal modal logic NormalNumber normal number NormalOfPlane normal of plane NormalOrder normal order NormalordinalFunction normal (ordinal) function NormalRandomVariable normal random variable NormalSection