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

TL;DR

This paper clarifies and corrects misconceptions in the theory of covering spaces and pseudo-covering spaces within digital topology, addressing errors and redundancies in previous literature.

Contribution

It provides corrected definitions, clarifies the relationships between covering and pseudo-covering spaces, and rectifies inaccuracies in prior works by Boxer, Han, and others.

Findings

Corrected and clarified definitions of covering and pseudo-covering spaces

Identified and fixed errors in previous literature on digital topology

Streamlined the relationships between different types of covering spaces

Abstract

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.