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**The adherences of languages as topological spaces.**
*(English)*
Zbl 0571.68057

Automata on infinite words, Ec. Printemps Inf. Théor., Le Mont Dore 1984, Lect. Notes Comput. Sci. 192, 147-163 (1985).

[For the entire collection see Zbl 0563.00019.]

The problem of characterizing the topological spaces that arise as adherences of languages of specified types is raised and pertinent concepts of general topology are reviewed. It is observed that the spaces that arise as adherences of arbitrary languages may be characterized as either: (1) the closed subsets of the Cantor ternary set; (2) the zero- dimensional compact metrizable spaces; or (3) the Stone spaces of the countable Boolean algebras. R. S. Pierce’s concept of a space of finite type is reviewed and his theorem characterizing the zero-dimensional compact metric spaces of finite type by means of an associated finite structural invariant is reviewed. It is shown that a topological space is homeomorphic with the adherence of a regular language if and only if it is zero-dimensional compact metrizable and of finite type. The structural invariant of the adherence of a regular language is algorithmically constructible from any automaton recognizing the language. Comparing these invariants provides a procedure for deciding homeomorphism of adherences for regular languages.

The problem of characterizing the topological spaces that arise as adherences of languages of specified types is raised and pertinent concepts of general topology are reviewed. It is observed that the spaces that arise as adherences of arbitrary languages may be characterized as either: (1) the closed subsets of the Cantor ternary set; (2) the zero- dimensional compact metrizable spaces; or (3) the Stone spaces of the countable Boolean algebras. R. S. Pierce’s concept of a space of finite type is reviewed and his theorem characterizing the zero-dimensional compact metric spaces of finite type by means of an associated finite structural invariant is reviewed. It is shown that a topological space is homeomorphic with the adherence of a regular language if and only if it is zero-dimensional compact metrizable and of finite type. The structural invariant of the adherence of a regular language is algorithmically constructible from any automaton recognizing the language. Comparing these invariants provides a procedure for deciding homeomorphism of adherences for regular languages.

### MSC:

68Q45 | Formal languages and automata |

54H99 | Connections of general topology with other structures, applications |

54A05 | Topological spaces and generalizations (closure spaces, etc.) |