A Jordan curve theorem with respect to a pretopology on ℤ²

Abstract

We study a pretopology on ℤ² having the property that the Khalimsky topology is one of its quotient pretopologies. Using this fact, we prove an analogue of the Jordan curve theorem for this pretopology, thus showing that such a pretopology provides a large variety of digital Jordan curves. Some consequences of this result are discussed too.

Keywords:

• quasi-discrete pretopology
• quotient pretopology
• connectedness graph
• digital plane
• Jordan curve

2010 AMS Subject Classifications: :

• Primary: 54D05
• 54A05
• 54B15
• Secondary: 52C99
• 05C40

Acknowledgements

This work was supported by the IT4Innovations Centre of Excellence project CZ.1.05/1.1.00/02.0070 and by the Brno University of Technology Grant Agency, project no. FSI-S-11-3.