alternative characterizations of Noetherian topological spaces, proof of

We prove the equivalence of the following five conditions for a topological spaceMathworldPlanetmath X:


Let f:P(X)P(X) be the complementPlanetmathPlanetmath map, i.e., f(A)=XA for any subset A of X. Then f induces an order-reversing bijectiveMathworldPlanetmathPlanetmath map between the open subsets of X and the closed subsets of X. Sending arbitrary chains ( of open/closed subsets of X through f then immediately yields the equivalence of conditions of the theorem, and also the equivalence of (Min) and (Max).

(Min) (DCC) is obvious, since the elements of an infiniteMathworldPlanetmath strictly descending chain of closed subsets of X would form a set of closed subsets of X without minimal element. Likewise, given a non-empty set S consisting of closed subsets of X without minimal element, one can construct an infinite strictly descending chain in S simply by starting with any A0S and choosing for An+1 any proper subsetMathworldPlanetmathPlanetmath of An satisfying An+1S. Hence, we have proven conditions (ACC), (DCC), (Min) and (Max) of the theorem to be equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, and will be done if we can prove equivalence of statement (HC) to the others.

To this end, first assume statement (HC), and let (Ui)i be an ascending sequencePlanetmathPlanetmath of open subsets of X. Then obviously, the Ui form an open cover of U=iUi, which by assumptionPlanetmathPlanetmath is bound to have a finite subcover. Hence, there exists n such that i=0nUi=iUi, so our ascending sequence is in fact stationary. Conversely, assume statement (Max) of the theorem, let AX be any subset of X and let (Ui)iI be a family of open sets in X such that the UiA form an open cover of A with respect to the subspace topology. Then, by the assumption, the set of finite unions of the Ui has at least one maximal element, say U, and with any iI we obtain UiU=U because of maximality of U. Hence, we have UiU for all iI, so in fact iIUi=U. But, U was a union of a finite number of Ui by construction; hence, a finite subcovering of U and thereby of A has been found. ∎

Title alternative characterizations of Noetherian topological spaces, proof of
Canonical name AlternativeCharacterizationsOfNoetherianTopologicalSpacesProofOf
Date of creation 2013-03-22 15:25:39
Last modified on 2013-03-22 15:25:39
Owner yark (2760)
Last modified by yark (2760)
Numerical id 12
Author yark (2760)
Entry type Proof
Classification msc 14A10