In this paper, we define three classes of languages of bi-infinite words, namely local bi-ω languages, recognizable bi-ω languages and Büchi local bi-ω languages as subclasses of the class of regular bi-ω languages and prove some basic results. We observe that the class of recognizable bi-ω languages coincides with the well-known class of rational bi-adherence languages and show that the class of regular bi-ω languages is the class of morphic images of Büchi local bi-ω languages. We provide learning algorithms for Büchi local bi-ω languages and regular bi-ω languages.