MINIMAL CLOZ-COVERS AND BOOLEAN ALGEBRAS
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In this paper, we first show that for any space X, there is a Boolean subalgebra G(zX) of R(X) containg G(X). Let X be a strongly zero-dimensional space such that z(b-1) (X) is the minimal cloz-coevr of X, where (Ecc(BX); z ) is the minimal cloz-cover of BX. We show that the minimal cloz-cover Ecc(X) of X is a subspace of the Stone space S(G(zX)) of G(zX) and that Ecc(X) is a strongly zero-dimensional space if and only if Ecc(X) and S(G(zX)) are homeomorphic. Using these, we show that Ecc(X) is a strongly zero-dimensional space and G(zX) = G(X) if and only if B Ecc(X) =Ecc(BX).
Abstract
In this paper, we first show that for any space X, there is a Boolean subalgebra G(zX) of R(X) containg G(X). Let X be a strongly zero-dimensional space such that z(b-1) (X) is the minimal cloz-coevr of X, where (Ecc(BX); z ) is the minimal cloz-cover of BX. We show that the minimal cloz-cover Ecc(X) of X is a subspace of the Stone space S(G(zX)) of G(zX) and that Ecc(X) is a strongly zero-dimensional space if and only if Ecc(X) and S(G(zX)) are homeomorphic. Using these, we show that Ecc(X) is a strongly zero-dimensional space and G(zX) = G(X) if and only if B Ecc(X) =Ecc(BX).
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