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The Eleventh Summer Conference on General Topology and Applications
August 10-13, 1995
University of Southern Maine
Gorham, ME, USA

Organizers
J. Baumgartner, D. Briggs, J. deBakker, B. Flagg, G. Gruenhage, M. Guay, Y. Kong, R. Kopperman, S. Shore, J. Rutten, J. Vaughan

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Path-Connected Sets in Digital Spaces
by
Gerhard X. Ritter
Unversity of Florida

It is well known that if S subset or equal Z2, where Z2 is the digital space with the von-Neumann topology, then the following are equivalent:

  1. S is 4-connecled,

  2. S is connected,

  3. S is path-connected, and

  4. S is digital path-connected.
It is also well known that there does not exist a topology on Z2 for which connectivily is equivalent to 8-connectivity. In this paper we review some old and provide several new results concerned with path-connectivity in digital spaces (i.e., Z2 together with a topology). Specifically, we investigate the properties of digital arcs, digital simple closed curves, digital Jordan arcs, and digital Jordan curves in different digital spaces. The notions of weak connectivity, weak path-connectivity, and shift invariant weak paths are explored. A particular consequence of this exploration shows that S subset or equal Z2 is an 8-path if and only if S is a shift invariant weak path.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaf-67.