The topological structure of the adherences of regular languages is considered using zero-dimensional compact metric spaces, studied by R. S. Pierce
[Mem. Am. Math. Soc. 130 (1972; Zbl 0253.54028
)]. It is shown that the adherence of any regular language L is of such finite type, and from any automaton recognizing L a finite invariant structure, called a structural diagram by the author, is algorithmically constructible. This result implies that homeomorphism of adherences is decidable for regular languages. It is also shown that every zero- dimensional compact metrizable space of finite type is homeomorphic with the adherence of a regular language, where the language can be chosen to be two-testable in the strict sense.