Self-Similar Topological Fractals

We introduce the notion of (abelian) similarity scheme, as a constructive model for topological self-similar fractals, in the same way in which the notion of iterated function system furnishes a constructive notion of self-similar fractals in a metric environment. At the same time, our notion gives a constructive approach to the Kigami-Kameyama notion of topological fractals, since a similarity scheme produces a topological fractal a la Kigami-Kameyama, and many Kigami-Kameyama topological fractals may be constructed via similarity schemes. Our scheme consists of objects X_n-> Y_n, <- X_n X Y_n, where X_n and Y_n are compact Hausdorff spaces, the first map is continuous injective and the second map is continuous surjective. This scheme produces a sequence (X_n) of compact Hausdorff spaces, X_n embedded in X_(n+1), and a compact Hausdorff space X_\infty giving a sort of injective limit space, which turns out to be self-similar. We observe that the space parametrizes the generalized similarity maps, and finiteness of is not required.