Topology Proceedings 24 Summer (1999).
Electronic version appeared 5 March 2001.
Abstract
We consider the problem of extrapolating the homology of a compact metric space from a finite point-set approximation. Our approach uses inverse systems of epsilon-neighborhoods and inclusion maps to derive relationships between the Betti numbers of the space and its approximation. The inclusion maps are necessary for the identification of topological features in an epsilon-neighborhood that persist in the limit as epsilon tends to zero. We outline an algorithm for a computational implementation. As an example, we present data for some iterated function system relatives of the Sierpinski triangle.
1991 AMS subject classification: Primary: 55-04; Secondary: 58F12, 28A80.
Keywords: Computational homology, Betti numbers, fractal geometry, shape theory.