Abstract
Combinatorics We discuss an Alexandroff topology on ℤ2 having the property that its quotient topologies include the Khalimsky and Marcus-Wyse topologies. We introduce a further quotient topology and prove a Jordan curve theorem for it.
Highlights
In the classical approach to digital topology, graph theoretic tools are used for structuring Z2, namely the well-known binary relations of 4-adjacency and 8-adjacency
We study another of its quotient topologies on Z2, denoted by v, and prove a Jordan curve theorem for it
Jordan curves play an important role in computer image processing because they represent boundaries of regions of digital images
Summary
In the classical approach to digital topology (see e.g. [12] and [13]), graph theoretic tools are used for structuring Z2, namely the well-known binary relations of 4-adjacency and 8-adjacency. Neither 4adjacency nor 8-adjacency itself allows an analogue of the Jordan curve theorem (cf [9]) and, 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 Z2, called the Khalimsky topology (cf [5]), for structuring the digital plane. We discuss a topology on Z2 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 Z2, denoted by v, and prove a Jordan curve theorem for it. Josef Slapal (ii) Those simple closed curves C in (Z2, v) to which the paper’s Jordan curve theorem applies will in most cases not be the common boundary of the two components of Z2 \ C
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