Essential concepts of digital topology (digital $k$-covering spaces and pseudo $k$-covering spaces)

The present paper focuses on the notions of covering spaces, pseudo-covering spaces, and their equivalences. We discuss something incorrectly mentioned in Boxer’s papers and correct them. Indeed, Sections 4-6 (or 4-6) of \cite{B3} are redundant because they have some incorrect assertions on $(k_1,k_2)$-covering spaces or pseudo- $(k_1,k_2)$-covering spaces due to his misunderstanding on Han’s papers \cite{H14,H16}. In addition, many things in \cite{B3} are duplicated with some results in \cite{H18}. In addition, since the papers \cite{P1,P2} also have some defects, we correct and improve them.