The Borel Structure of the Collections of Sub-Self-Similar Sets and Super-Self-Similar Sets

We show that the sets of sub-self-similar sets and super-self-similar sets are both dense, first category, F oe subsets of K(R d ), the Hausdorff metric space of non-empty compact, subsets of R d . We also investigate the set of self-similar sets as a subset of the sub-self-similar sets and the super-self-similar sets. 1 Introduction In [Fal1], Falconer introduced the notion of sub-self-similarity as a generalization of self-similarity and showed that sub-self-similar sets retain many of the nice properties of self-similar sets. Later in [Fal2] we find the notion of a superself -similar set. The question arises as to how strong a generalization are these new concepts. In this paper, we quantify this question using topological notions in K(R d ), the Hausdorff metric space of non-empty compact subsets of R d . In particular, we show that the sets of sub-self-similar sets and super-self-similar sets are both dense, first category, F oe subsets of K(R d ). The fact that these s..