Summary: We consider the problem of extrapolating the homology of a compact metric space from a finite point-set approximation. Our approach is based on inverse systems of \(\varepsilon\)-neighborhoods and inclusion maps. We show that the inclusion maps are necessary to identify topological features in an \(\varepsilon\)-neighborhood that persist in the limit as \(\varepsilon\to 0\). We outline a possible algorithm for computer implementation. As test examples, we present data for some iterated function system relatives of the Sierpinski triangle.