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The notions of a local $(k_0,k_1)$-isomorphism and a weakly local$(k_0,k_1)$-isomorphism play crucial roles in developing a digital$(k_0,k_1)$-covering space and a pseudo-$(k_0,k_1)$-covering space,respectively. In relation to the study of pseudo-$(k_0,k_1)$-covering spaces,since there are some works to be refined and improved in the literature, therecent paper \cite{H10} improved and corrected some mistakes occurred in theliterature. One of the important things is that the notion of apseudo-$(k_0,k_1)$-covering map in \cite{H6,H9} was revised to be morebroadened in \cite{H10}. Thus this new version is proved to be equivalent to aweakly local $(k_0,k_1)$-isomorphic surjection \cite{H10}. The present papercontains some works in \cite{H10} and we only deals with $k$-connected digitalimages $(X, k)$.

Pseudocovering and digital covering spaces