Compactness of certain bounded zero-sets in completely regular spaces

Let X be a completely regular T 1 -space. A zero-set Z in X is called a full zero-set if cl β X Z is a zero-set in the Čech–Stone compactification βX of X. As an generalization of theorems by W.G. McArthur and V. V. Uspenskii, we prove that every bounded, full zero-set F in X is compact if either (i) X has a regular G δ -diagonal or (ii) X is a Baire space such that every open cover has a σ-point-finite open refinement. In case (i), F is metrizable by Šneıˇderʹs theorem. We also apply this to show that if the Dieudonné completion μX of X is a paracompact M-space, then X is metrizable if either (i) or (iii) X is a Baire space with a σ-point-finite base.