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A Stone's Representation Theorem And Some Applications, Eissa D. Habil

Turkish Journal of Mathematics

In this paper, we prove the following form of Stone's representation theorem: Let \sum be a \sigma-algebra of subsets of a set X. Then there exists a totally disconnected compact Hausdorff space {\cal K} for which (\sum, \cup, \cap) and ({\cal C}({\cal K}), \cup ,\cap), where {\cal C}({\cal K}) denotes the set of all clopen subsets of {\cal K}, are isomorphic as Boolean algebras. Furthermore, by defining appropriate joins and meets of countable families in {\cal C}({\cal K}), we show that such an isomorphism preserves \sigma-completeness. Then, as a consequence of this result, we obtain the result that if ba(X,\sum) …