Dimension for Alexandrov spaces
Petra Wiederhold, Richard G. Wilson · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1993
This paper continues the study of the topological model of the support of a digital image published by Kronheimer in 1992. There, he interpreted the generation of the support D of the image from a topological space S by means of some 'discretization' as the construction of a quotient space (Delta) of S, which represents the set D an d has a reasonable (non-discrete) topology. Under some conditions the space (Delta) is an Alexandrov space. Having in mind the practical example S equals Rn and D equals Zn we speak of 'n-dimensional images', although there is no dimension on the space (Delta) . We define in this paper a so-called Alexandrov dimension for arbitrary Alexandrov spaces. Under this definition an image which was sampled from a function defined on Rn has dimension n. If the Alexandrov space (Delta) is T0, then it corresponds to a canonical partially ordered set ((Delta) , ≤). We prove, that in this case the Alexandrov dimension coincides with the height of ((Delta) , ≤).