  The topological and metrical equivalence of fractals is an important topic in
analysis. In this paper, we use a class of finite state automata, called
$\Sigma$-automaton, to construct psuedo-metric spaces, and then apply them to
the study of classification of self-similar sets. We first introduce a notion
of topology automaton of a fractal gasket, which is a simplified version of
neighbor automaton; we show that a fractal gasket is homeomorphic to the
psuedo-metric space induced by the topology automaton. Then we construct a
universal map to show that psuedo-metric spaces induced by different automata
can be bi-Lipschitz equivalent. As an application, we obtain a rather general
sufficient condition for two fractal gaskets to be homeomorphic or Lipschitz
equivalent.
