Title

Spaces With a Pretty Base

Document Type

Article

Department

Mathematics (HMC)

Publication Date

3-15-1991

Abstract

Throughout, R will denote a commutative ring whose only nilpotent element is 0 (i.e., R is reduced) and mR its set of minimal prime ideals. If S is a subset of R, let A(S)={xeR:xS=O}, let h(S)={PE~R:SCP}, and let hc(S)=mR\h(S). If S Є R is a singleton, abbreviate A((s}) by A(s), h({s}) by h(s), and hc({s}) by hc(s). It is well known that for any S Є R, hc(s) =h(A(s)); see, for example [.5]. Unless noted otherwise, mR will carry the hull-kernel topology t; i.e., the topology whose base is {hC(.s):sЄR}.

It is shown in [5] that (mR, T) is a zero-dimensional Hausdorff space. In the sequel, it will be assumed also that R satisfies: (CAC) {A(s): s Є R} is closed under countable intersection. It is shown in [5] and [6] that if R is either the ring of all continuous real-valued functions on a topological space or the ring of formal power series over a commutative reduced ring, then R satisfies CAC.

In [3] it was shown that if R satisfies CAC, and S⊂mR is weakly Lindelijf (i.e., every open cover of S has a countable subfamily whose union is dense in S), then Clmr S and the Stone-Čech compactification βS of S are homeomorphic. That is, (i) S is C*-embedded in its closure, and (ii) Clmr S is compact. (Notation and terminology unfamiliar to the reader may be found in [4].)

(Recall from [l] that spaces of countable cellularity as well as Lindelof spaces are weakly Lindelof, while an uncountable discrete space is not.)

The proof of these assertions about mR in [3] involves a mixture of algebraic and topological techniques. In this note, the algebraic assumptions needed to deduce (i) and (ii) are converted into topological assumptions from which it is possible to prove them in a purely topological way. Some results new even for (mR, τ) are given as well. These results were obtained jointly with R. Kopperman and R.G. Woods, and the three of us have developed similar techniques that enable us to obtain results about other classes of spaces including subspaces of F-spaces.

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©1991 Elsevier

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