Topological structuring of the digital plane

In the classical approach to digital topology (see e.g. [12] and [13]), graph theoretic tools are used for structuring Z, namely the well-known binary relations of 4-adjacency and 8-adjacency. But neither 4adjacency nor 8-adjacency itself allows an analogue of the Jordan curve theorem (cf. [9]) and, therefore, one has to use a combination of the two adjacencies. To overcome this disadvantage, a new, purely topological approach to the problem was proposed in [6] which utilizes a convenient topology on Z, called the Khalimsky topology (cf. [5]), for structuring the digital plane. At present, this topology is one of the most important concepts of digital topology. It has been studied and used by many authors, see e.g. [3] and [7]-[10]. The possibility of employing convenient topological structures on Z different from the Khalimsky topology is discussed in [14]-[19]. Particularly, in [16], a new topology on Z is introduced and it is shown there that this topology provides certain convenient Jordan curves behaving more advantageously than the Jordan curves in the Khalimsky space. The quotient topologies of the topology from [16] are studied in [17] where it is shown that they include, among others, the Khalimsky and Marcus-Wyse topologies. In the present note we continue the investigations from [16] and [17]. We discuss a topology on Z which is finer than the topology introduced in [16] but still has the property that the Khalimsky and Marcus-Wyse topologies belong to its quotient topologies. We study another of its quotient topologies on