Minimal Cloz-Covers and Boolean Algebras

Abstract

Abstract. In this paper, we first show that for any space X, there is a Boolean subalgebra G(z X ) of R(X) containg G(X). Let X be a strongly zero-dimensional space such that z −1 β (X) is the minimal cloz-coevr of X, where (E cc (βX), z β ) is the minimal cloz-cover of βX. We show that the minimal cloz-cover E cc (X) of X is a subspace of the Stone space S(G(z X )) of G(z X ) and that E cc (X) is a strongly zero-dimensional space if and only if βE cc (X) and S(G(z X )) are homeomorphic. Using these, we show that E cc (X) is a strongly zero-dimensional space and G(z X ) = G(X) if and only if βE cc (X) = E cc (βX).

Minimal Cloz-Covers of Κx

“…Then (E cc (X), z X ) is the minimal cloz-cover of X, where z X ((α, x)) = x( [3]). It was shown that every space has a minimal cloz-cover( [7]).…”

“…Then E cc (βX) = S(G(βX))( [3])and E cc (βX) is a zero-dimensional space. Since z −1 β (kX) is the minimal cloz-cover of kX, βE cc (kX) and S(z k (G(T ))) are homeomorphic ( [7]). By (3), S(z k (G(T ))) and S(G(kX)) are homeomorphic.…”