# Can any c.e. language with infinite words be decomposed into infinite CFLs with infinite words?

Suppose $L$ is a computably enumerable language, can it be decompose into infinite CFLs with infinite words ? $$L=\bigcup_{L_i\in CFL }^{\infty}L_i$$

Second question: if it is possible that every $L$ , computably enumerable language with infinity words be decompose into finite/infinite CFLs with infinite words, can it be be decompose into finite/infinite unabiguous CFLs with infinite words?

UpDate: Third question, is it possible that every $L$ , computably enumerable language with infinity words be formed by union,complement,intersection,product (finite/infinite) of unabiguous CFLs with infinite words?

No, we get counterexamples by considering resource-bounded randomness. In fact the gap between c.e. and $\mathsf{CFL}$ is wide.
Let $R$ be exponential-time random. Then $R$ is $\mathsf{NP}$-immune, i.e., it has no infinite subset in $\mathsf{NP}$. In particular it has no infinite subset that is a context-free language.