Abstract
In this paper, we first show that for any space X, there is a <TEX>${\sigma}$</TEX>-complete Boolean subalgebra of <TEX>$\mathcal{R}$</TEX>(X) and that the subspace {<TEX>${\alpha}{\mid}{\alpha}$</TEX> is a fixed <TEX>${\sigma}Z(X)^{\sharp}$</TEX>-ultrafilter} of the Stone-space <TEX>$S(Z({\Lambda}_X)^{\sharp})$</TEX> is the minimal basically disconnected cover of X. Using this, we will show that for any countably locally weakly Lindel<TEX>$\ddot{o}$</TEX>f space X, the set {<TEX>$M{\mid}M$</TEX> is a <TEX>${\sigma}$</TEX>-complete Boolean subalgebra of <TEX>$\mathcal{R}$</TEX>(X) containing <TEX>$Z(X)^{\sharp}$</TEX> and <TEX>$s_M^{-1}(X)$</TEX> is basically disconnected}, when partially ordered by inclusion, becomes a complete lattice.
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