alternative characterizations of Noetherian topological spaces

Let X be a topological spaceMathworldPlanetmath. The following conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath conditions for X to be a Noetherian topological space:

  1. 1.

    X satisfies the descending chain conditionMathworldPlanetmathPlanetmathPlanetmath ( for closed subsets.

  2. 2.

    X satisfies the ascending chain conditionMathworldPlanetmathPlanetmath ( 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 compactPlanetmathPlanetmath.

Title alternative characterizations of Noetherian topological spaces
Canonical name AlternativeCharacterizationsOfNoetherianTopologicalSpaces
Date of creation 2013-03-22 14:16:25
Last modified on 2013-03-22 14:16:25
Owner yark (2760)
Last modified by yark (2760)
Numerical id 10
Author yark (2760)
Entry type Theorem
Classification msc 14A10