Annali di Matematica Pura ed Applicata

, Volume 126, Issue 1, pp 253–266 | Cite as

Lo spazio duale di un prodotto di algebre di Boole e le compattificazioni di Stone

  • Claudio Bernardi


This paper is concerned with the Stone space X of a direct product\(B = \prod\limits_{i \in I} {B_i }\) of infinitely many Boolean algebras. In paragraph 2, after recalling that X is the Stone-Čech compactification of the sum (disjoint union)\(\sum\limits_{i \in I} {X_i }\) of the Stone spaces of the algebras Bi, we exhibit a compactification of\(\sum\limits_{i \in I} {X_i }\) which is not a Stone space and we give a method to construct all the «Stone compactifications» of\(\sum\limits_{i \in I} {X_i }\) (the corresponding Boolean algebras are easily characterised). In paragraph 3, a set of ultrafilters of B (the «decomposable» ultrafilters) are introduced: this set properly contains\(\sum\limits_{i \in I} {X_i }\), but, as is shown in paragraph 5, there are direct products that admit nondeeomposable ultrafilters (this is the case iff the set {Card Bi: i ε I } is not bounded by a natural number). In paragraph 4, among other things, we prove, for the set of decomposable ultrafilters, a weak form of countable compactness, in the sense that every countable clopen cover has a finite subcover; then, we deduce that the set of decomposable ultrafilters is pseudocompact, while obviously\(\sum\limits_{i \in I} {X_i }\) is not. Lastly, in paragraph 6, we give a second characterisation of the Stone space of B, showing that every ultrafilter of B can be obtained by iterating in a suitable way the procedure which leads to the construction of decomposable ultrafilters.


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© Fondazione Annali di Matematica Pura ed Applicata 1980

Authors and Affiliations

  • Claudio Bernardi
    • 1
  1. 1.Siena

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