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The Khalimsky Line as a Foundation for Digital Topology

  • Conference paper
Shape in Picture

Part of the book series: NATO ASI Series ((NATO ASI F,volume 126))

Abstract

An object is defined from which digital spaces can be built. It combines the “one-dimensional connectedness” of intervals of reals with a“point-bypoint” quality necessary for constructing algorithms, and thus serves as a foundation for digital topology. Ideas expressed in quotation marks here are given precise meanings. This study considers the Khalimsky line, that is, the integers, equipped with the topology in which a set is open iff whenever it contains an even integer, it also contains its adjacent integers. It is shown that this space and its interval subspaces are those satisfying the conditions mentioned previously. The Khalimsky line is used to study digital connectedness and homotopy.

The author wishes to acknowledge comments by David Foster, Paul Meyer, and two unknown referees, which led to substantial improvements in this paper.

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Author information

Authors and Affiliations

  1. Department of Mathematics, City College of New York, New York, NY, 10031, USA

    Ralph Kopperman

Authors
  1. Ralph Kopperman
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Editor information

Editors and Affiliations

  1. Centre for Mathematics and Computer Science (CWI), Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands

    Ying-Lie O  & Henk J. A. M. Heijmans  & 

  2. Institute for Human Factors (TNO-IZF), Netherlands Organization for Applied Scientific Research, Kampweg 5, 3769 DE, Soesterberg, The Netherlands

    Alexander Toet

  3. Department of Communication and Neuroscience, University of Keele, ST5 5BG, Keele, Staffordshire, UK

    David Foster

  4. Department of Electrical and Computer Engineering, Rutgers University, 08855-0909, Piscataway, NJ, USA

    Peter Meer

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© 1994 Springer-Verlag Berlin Heidelberg

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Kopperman, R. (1994). The Khalimsky Line as a Foundation for Digital Topology. In: O, YL., Toet, A., Foster, D., Heijmans, H.J.A.M., Meer, P. (eds) Shape in Picture. NATO ASI Series, vol 126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03039-4_2

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  • DOI: https://doi.org/10.1007/978-3-662-03039-4_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08188-0

  • Online ISBN: 978-3-662-03039-4

  • eBook Packages: Springer Book Archive

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