A technique in the topology of connected self-similar tiles

Not much is known about the topological structure of a connected self-similar tile whose interior is disconnected, and even less is understood if the interior consists of infinitely many components. We introduce a technique to show that for a large class of self-similar tiles in ℝ², the closure of each component of the interior is homeomorphic to a disk. This allows us to prove such a result for the Eisenstein set, the fundamental domain of a well-known quadratic canonical number system, and some other well-known fractal tiles.