The main question of this paper is to retrieve some continuity properties on (discrete) T0-Alexandroff spaces. One possible application, which will guide us, is the construction of the so-called “tree of shapes” (intuitively, the tree of level lines). This tree, which should allow to process maxima and minima in the same way, faces quite a number of theoretical difficulties that we propose to solve using set-valued analysis in a purely discrete setting. We also propose a way to interpret any function defined on a grid as a “continuous” function thanks to an interpolation scheme. The continuity properties are essential to obtain a quasi-linear algorithm for computing the tree of shapes in any dimension, which is exposed in a companion paper [10].


Topological Space Discrete Space Digital Topology Space Subdivision Classical Continuous Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Laurent Najman
    • 1
  • Thierry Géraud
    • 2
  1. 1.LIGM, Équipe A3SI, ESIEEUniversité Paris-EstFrance
  2. 2.EPITA Research and Development Laboratory (LRDE)France

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