MINIMAL CLOZ-COVERS OF NON-COMPACT SPACES

Observing that for any dense weakly Lindelof subspace of a space Y, X is $Z^{#}$ -embedded in Y, we show that for any weakly Lindelof space X, the minimal Cloz-cover ($E_{cc}$(X), $z_{X}$) of X is given by $E_{cc}$(X) = {(\alpha, \chi$) : $\alpha$ is a G(X)-ultrafilter on X with $\chi\in\cap\alpha$}, $z_{X}$=(($\alpha, \chi$))=$\chi$, $z_{X}$ is $Z^{#}$ -irreducible and $E_{cc}$(X) is a dense subspace of $E_{cc}$($\beta$X).