Let
X be a completely regular
Formula Not Shown -space. A zero-set
Z in
X is called a full zero-set if
Formula Not Shown is a zero-set in the
Cech–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
Formula Not Shown -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 Šne?
![](/images/glyphs/CAK.GIF)
der's 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.