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