Special bounded zero-sets and their applications

All spaces considered are completely regular $T_{1} $-spaces and all maps are continuous. For aspace $X $ , let $\mathcal{B}X $ denote the $\check{\mathrm{C}}\mathrm{e}\mathrm{c}\mathrm{h}$-Stone compactification of $X $ and $\mu X $ the Dieudonn\’e completion (i.e., the completion with respect to the finest uniformity) of $X $. For aspace X. $C(X) $ denotes the set of all real-valued, continuous functions on $X $ and asubset $A $ of $X $ is said to be bounded (or relatively $pseudocompact^{\backslash}$) in $X$ if every $f\in C(X) $ is bounded on $A $. A zerO-set in a space $X $ is a set of the form $f^{-1}(0) $ for so me $f\in C_{(}^{(}X $ ). We now call a zer0-set $Z $ in $X $ a full zerO-set if $\mathrm{c}1_{\beta X}Z $ is azer0-set in $\beta X $. Such azer0-set was studied by Rudd $\lceil 15_{\rfloor}^{\rceil} $ in the context of rings of continuous functions. In this paper, we study basic properties of full zer0-sets and consider the problem when abounded, full zer0-set is compact. In particular, we prove that: $\backslash /\mathrm{i}) $ Let $X $ be aspace with aregular $G_{\delta}$-diagonal. Then every bounded, full zer0-set in $X $ is compact and metrizable. (ii) Let $X $ be aBaire space such that every open cover has aa-point-finite open refinement. Then, every bounded, full zer0-set in $X $ is compact. (iii) Let $X $ be aBaire space with aa-point-finite base. Then, every bounded, full zer0-set in $X $ is compact and metrizable. $1^{\backslash}11\mathrm{e} $ above results are generalizations of the following theorems: (1) (McArthur [11]) Every pseudocompact space with aregular $G_{\delta} $-diagonal is metrizable. (2) (Uspenskii [16]) Every pseudocompact space such that every open cover has a cr-point-finite open refinement is compact. (3) (Uspenskii [16]) Every pseudocompact space with aa-point-finite base is metriz-able