  Finite automata were used to determine multiple addresses in number systems
and to find topological properties of self-affine tiles and finite type
fractals. We join these two lines of research by axiomatically defining
automata which generate topological spaces. Simple examples show the potential
of the concept. Spaces generated by automata are topologically self-similar.
Two basic algorithms are outlined. The first one determines automata for all
$k$-tuples of equivalent addresses from the automaton for double addresses. The
second one constructs finite topological spaces which approximate the generated
space. Finally, we discuss the realization of automata-generated spaces as
self-similar sets.
