Concept
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. The Noetherian property of a topological space can also be seen as a strong compactness condition, namely that every open subset of such a space is compact, and in fact it is equivalent to the seemingly stronger statement that every subset is compact. A topological space is called Noetherian if it satisfies the descending chain condition for closed subsets: for any sequence of closed subsets of , there is an integer such that A topological space is Noetherian if and only if every subspace of is compact (i.e., is hereditarily compact), and if and only if every open subset of is compact. Every subspace of a Noetherian space is Noetherian. The continuous image of a Noetherian space is Noetherian. A finite union of Noetherian subspaces of a topological space is Noetherian. Every Hausdorff Noetherian space is finite with the discrete topology. Proof: Every subset of X is compact in a Hausdorff space, hence closed. So X has the discrete topology, and being compact, it must be finite. Every Noetherian space X has a finite number of irreducible components. If the irreducible components are , then , and none of the components is contained in the union of the other components. Many examples of Noetherian topological spaces come from algebraic geometry, where for the Zariski topology an irreducible set has the intuitive property that any closed proper subset has smaller dimension. Since dimension can only 'jump down' a finite number of times, and algebraic sets are made up of finite unions of irreducible sets, descending chains of Zariski closed sets must eventually be constant. A more algebraic way to see this is that the associated ideals defining algebraic sets must satisfy the ascending chain condition.