Topology Proceedings 28 No. 2 (2004), pp. 487-581

On sequential properties of Noetherian topological spaces

Ivan Gotchev and Hristo Minchev

Sequential properties of Noetherian topological spaces are considered. A topological space X is called Noetherian if for every increasing by inclusion sequence (U_n)_n=1^¥ of open subsets of X there exists n such that U_n = U_n+1 = ¼. It is shown that every Noetherian topological space is sequentially compact and that the sequential topology inherits the Noetherian property. Hence, every sequentially open cover of a Noetherian topological space has a finite subcover. The following result is proved: Let X be a Noetherian topological space in which every irreducible closed subset F has a generic point. The space X is sequential if and only if h(X) £ w₁, where h(X) is a suitable ordinal invariant. From this result follows that a Zariski space X is sequential if and only if h(X) £ w₁ and that if R is a commutative Noetherian ring then the prime spectrum Spec R is a sequential Noetherian topological space.

Keywords: Noetherian ring, Noetherian topological space, Zariski space, sequentially compact, s-compact, compact, sequential space

Mathematics Subject Classification: 54D55 54D20 13E05 (54D30 54D10 54A10)

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Topology Proceedings, Volume 28, No. 2 (2004)
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