On sequential properties of Noetherian topological spaces. (English) Zbl 1086.54015
Let \((X,\tau)\) be a Noetherian topological space (every strictly ascending chain of open sets is finite). The authors show that \((X,\tau)\) is sequentially compact and \(s\)-compact (related notions of semicompactness are discussed). Further, they define \(h(x)\), the height of \(X\) (related notions of dimension are discussed) and show that, if every irreducible closed subset of \(X\) has a generic point, then \(X\) is sequential if and only if \(h(X)\leq\omega_1\); this implies that the prime spectrum of a commutative Noetherian ring is a sequential Noetherian topological space.
Reviewer: Bernhard Behrens (Göteborg)
MSC:
54D55 | Sequential spaces |
54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |
13E05 | Commutative Noetherian rings and modules |
54D30 | Compactness |
54D10 | Lower separation axioms (\(T_0\)–\(T_3\), etc.) |
54A10 | Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) |
13J99 | Topological rings and modules |