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On Sequential Properties of Noetherian Topological Spaces
by
Ivan Gotchev
American University in Bulgaria
Coauthors: Hristo Minchev (University of Forestry and Wood Engineering)
A topological space X is called Noetherian if every decreasing by inclusion sequence {Fn}n=1\infty of closed subsets of X is stationary, i.e. there exists n such that Fn = Fn+1 = ... .
Let R be a commutative ring. SpecR is the set of all prime ideals of R provided with the topology in which F subset SpecR is closed if and only if there exists an ideal I of R such that F = { p in SpecR | p contains or equal I }.
A commutative ring R is called Noetherian if every increasing by inclusion sequence of ideals of R is stationary.
It is well known [1] that: (a) Every Noetherian topological space is compact. (b) Every subspace of a Noetherian topological space is Noetherian. (c) A topological space X is Noetherian if and only if every open subset of X is compact. (d) If R is a Noetherian commutative ring then SpecR is a Noetherian topological space.
A cover of a topological space X is called sequentially open cover if its elements are sequentially open sets.
A topological space X is s-compact [2] if every sequentially open cover of X has a finite subcover.
For a topological space X the following conditions are equivalent [2]:
(a) Every countable sequentially open cover of X has a finite subcover.
(b) X is a sequentially compact space.
It is known [2] that: (a) For the class of sequential spaces s-compactness coincides with compactness. (b) Every s-compact space is a compact space. (c) Every s-compact space is a sequentially compact space.
There exists an example of a compact and sequentially compact space, which is not s-compact [2].
Theorem 1 Every Noetherian topological space is s-compact and thus it is sequentially compact.
A topological space X is called irreducible [1] if the intersection of any finite number of open nonempty sets in X is nonempty.
Let X be a Noetherian topological space and P be the set of all closed irreducible subsets of X , ordered by inclusion. Let \alpha be the supremum of all ordinals \beta such that there exists strictly increasing function f : [0, \beta] --> P. It is called that the height of X is \alpha (h(X) = \alpha) if \alpha is infinite ordinal and h(X) = \alpha- 1 if \alpha is finite.
There exists an example of a countable Noetherian topological space Y such that h(Y) = 2 and which is not sequential.
Theorem 2 Let X be a Noetherian topological space in which every irreducible subset F has common (general) point x, i.e. [`{x}] = F. The space X is sequential if and only if h(X) <= \omega1.
Corollary Let R be a commutative Noetherian ring. Then the prime spectrum SpecR is a sequential Noetherian topological space.
Date received: June 24, 1996
Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-24.