Digital covering spaces, Pseudo covering spaces, and related works.

The paper aims to find a condition which supports an equivalence among a local (k 0 , k 1) -isomorphism, a local (k 0 , k 1) -isomorphic surjection, and a (k 0 , k 1) -covering map. Indeed this work still remains open. To make a success of addressing this issue, the paper first compares among a digital-topological (D T -, for brevity) k -embedding, a local (L -, for short) k -isomorphism, a weakly local (W L -, for simplicity) k -isomorphic surjection, a (k 0 , k 1) -covering map, and a pseudo- k -covering map and further, deals with some related works. Second, assume a function f : (X , k 0) → (Y , k 1). Then we point out that the k 1 -connectedness of (Y , k 1) plays a crucial role in characterizing some relationships between an L - (k 0 , k 1) -isomorphism and a (k 0 , k 1) -covering map. Besides, a (k 0 , k 1) -covering map is proved to be stronger than a new version of the pseudo- (k 0 , k 1) -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 L - (k 0 , k 1) -isomorphism and a pseudo- (k 0 , k 1) -covering map. Finally, the map p : S C k n , l ′ → S C k n , l above is an L - k -isomorphism if and only if l ′ = t l , t ∈ ℕ. [ABSTRACT FROM AUTHOR]