On the dimensional properties of Stone-Čech remainder of $P_ 0$-spaces

Abstract:

A space X is called a ${P_0}$-space if there exists a perfect mapping f from X onto a metric space Y such that $\dim f = \sup \{ {f^{ - 1}}(y):y \in Y\} = 0$. We prove that the ${P_0}$-space X is almost weakly infinite dimensional iff the remainder $\beta X\backslash X$ of the Stone-Čech compactification $\beta X$ of X is A-weakly infinite dimensional. Furthermore we prove that $\Delta (\beta X\backslash X = {\text {ind}}(\beta X\backslash X) = {\text {Ind}}(\beta X\backslash X) = \dim (\beta X\backslash X)$ for the ${P_0}$-space X.