The authors study and compare several mechanisms for defining sets of biinfinite words (ωω-languages), namely ωω-finite automata, adherences of regular languages and sofic systems. They show that a sofic system is a topologically closed ωω-regular set. The authors also show that there is a one-to-one correspondence between sofic systems and the adherences of regular languages. They give a complete proof of the closure of the ωω-regular sets under ωω-rational relations and under Boolean operations. Finally, the authors disprove Hurd’s conjecture on bi-extensible subsets of languages, and show that the conjecture would hold if a different definition were used.
