In this paper, we construct a cover (L(X), cx) of a space X such that for any cloz-cover (Y, f) of X, there is a covering map g:Y→L(X) with cxog=f. Using this, we show that every Tychonoff space X has a minimal cloz-cover (Ecc(X), zx) and that for a strongly zero-dimensional space X, βEcc(X)=Ecc(βX) if and only if Ecc(X) is z#-embedded in Ecc(βX).