BASICALLY DISCONNECTED COVERS OF THE EXTENSION κX OF A SPACE X

Abstract

Observing that every Tychonoff space X has a weakly Lindel<TEX>$\ddot{o}$</TEX>f extension <TEX>${\kappa}X$</TEX> and the minimal basically diconneted cover <TEX>${\Lambda}{\kappa}X$</TEX> of <TEX>${\kappa}X$</TEX> is weakly Lindel<TEX>$\ddot{o}$</TEX>f, we first show that <TEX>${\Lambda}_{{\kappa}X}:{\Lambda}{\kappa}X{\rightarrow}{\kappa}X$</TEX> is a <TEX>$z^{\sharp}$</TEX>-irreducible map and that <TEX>${\Lambda}{\beta}X={\beta}{\Lambda}{\kappa}X$</TEX>. And we show that <TEX>${\kappa}{\Lambda}X={\Lambda}{\kappa}X$</TEX> if and only if <TEX>${\Lambda}^{\kappa}_X:{\kappa}{\Lambda}X{\rightarrow}{\kappa}X$</TEX> is an onto map and <TEX>${\beta}{\Lambda}X={\Lambda}{\beta}X$</TEX>.