In this paper, we prove that the correspondences $\underline{V}\Rightarrow V^{\omega},$ $\underline{V}\Rightarrow \vec{V}$, proposed by D. Perrin (1982) between $M$-varieties $\underline{V}$'s of finite monoids and varieties of recognizable omega-languages $V^{\omega}$'s and $\vec{V}$'s are one-to-one. New definitions of saturation and syntactic monoid of adherences of $\omega$-languages basing on the limit operation are introduced. As consequence, a new type of varieties generated by adherences of $\omega$-languages is defined and studied.