Digital covering spaces, Pseudo covering spaces, and related works

The paper aims to find a condition which supports an equivalence among a local [Formula: see text]-isomorphism, a local [Formula: see text]-isomorphic surjection, and a [Formula: see text]-covering map. Indeed this work still remains open. To make a success of addressing this issue, the paper first compares among a digital-topological ([Formula: see text]-, for brevity) [Formula: see text]-embedding, a local ([Formula: see text]-, for short) [Formula: see text]-isomorphism, a weakly local ([Formula: see text]-, for simplicity) [Formula: see text]-isomorphic surjection, a [Formula: see text]-covering map, and a pseudo-[Formula: see text]-covering map and further, deals with some related works. Second, assume a function [Formula: see text]. Then we point out that the [Formula: see text]-connectedness of [Formula: see text] plays a crucial role in characterizing some relationships between an [Formula: see text]-[Formula: see text]-isomorphism and a [Formula: see text]-covering map. Besides, a [Formula: see text]-covering map is proved to be stronger than a new version of the pseudo-[Formula: see text]-covering map in [S.-E. Han, Remarks on pseudocovering spaces in a digital topological setting, Filomat 38(2) (2024) 1–9]. Third, we propose a condition supporting the product property of an [Formula: see text]-[Formula: see text]-isomorphism and a pseudo-[Formula: see text]-covering map. Finally, the map [Formula: see text] above is an [Formula: see text]-[Formula: see text]-isomorphism if and only if [Formula: see text].