In this paper, we first show that $z_{{kappa}X}:E_{cc}({kappa}X){ ightarrow}{kappa}X$ is $z^{sharp}$-irreducible and that if $mathcal{G}(E_{cc}({eta}X))$ is a base for closed sets in ${eta}X$, then $E_{cc}({kappa}X)$ is $C^*$-embedded in $E_{cc}({eta}X)$, where ${kappa}X$ is the extension of X such that $vX{subseteq}{kappa}X{subseteq}{eta}X$ and ${kappa}X$ is weakly Lindel$ddot{o}$f. Using these, we will show that if $mathcal{G}({eta}X)$ is a base for closed sets in ${eta}X$ and for any weakly Lindel$ddot{o}$f space Y with $X{subseteq}Y{subseteq}{kappa}X$, ${kappa}X=Y$, then $kE_{cc}(X)=E_{cc}({kappa}X)$ if and only if ${eta}E_{cc}(X)=E_{cc}({eta}X)$.