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 X0​↪φ​X1​←πY×X0​, where X0​,X1​ and Y are compact Hausdorff spaces, the map φ is continuous injective and the map π is continuous surjective. This scheme produces a sequence Xn​, n∈N, of compact Hausdorff spaces, Xn​ embedded in Xn+1​, and a compact Hausdorff space X∞​ giving a sort of injective limit space, which turns out to be self-similar. We observe that the space Y parametrises the generalised similarity maps, and finiteness of Y is not required.