The Most Refined Axiom for a Digital Covering Space and Its Utilities

Sang-Eon Han
Additional contact information
Sang-Eon Han: Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju-City Jeonbuk 54896, Korea

Mathematics, 2020, vol. 8, issue 11, 1-21

Abstract: This paper is devoted to establishing the most refined axiom for a digital covering space which remains open. The crucial step in making our approach is to simplify the notions of several types of earlier versions of local ( k 0 , k 1 ) -isomorphisms and use the most simplified local ( k 0 , k 1 ) -isomorphism. This approach is indeed a key step to make the axioms for a digital covering space very refined. In this paper, the most refined local ( k 0 , k 1 ) -isomorphism is proved to be a ( k 0 , k 1 ) -covering map, which implies that the earlier axioms for a digital covering space are significantly simplified with one axiom. This finding facilitates the calculations of digital fundamental groups of digital images using the unique lifting property and the homotopy lifting theorem. In addition, consider a simple closed k : = k ( t , n ) -curve with five elements in Z n , denoted by S C k n , 5 . After introducing the notion of digital topological imbedding, we investigate some properties of S C k n , 5 , where k : = k ( t , n ) , 3 ≤ t ≤ n . Since S C k n , 5 is the minimal and simple closed k -curve with odd elements in Z n which is not k -contractible, we strongly study some properties of it associated with generalized digital wedges from the viewpoint of fixed point theory. Finally, after introducing the notion of generalized digital wedge, we further address some issues which remain open. The present paper only deals with k -connected digital images.

Keywords: local ( k 0 , k 1 )-isomorphism; unique lifting property; homotopy lifting theorem; digital covering; digital topological imbedding; generalized digital wedge (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/11/1868/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/11/1868/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2020:i:11:p:1868-:d:435376

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().