subsets of non regular language I know that there are many languages that are context free but not regular
like $\{a^n b^n :n>0\}.$
But I want to know if every context free but non-regular language has infinitely many non-regular subsets?
Thank you. 

If $L$ is context-free but not regular, $L$ must be infinite. For each $w\in L$ let $L_w=L\setminus\{w\}$. The finite language $\{w\}$ is certainly regular, and the union of two regular languages is regular, so if $L_w$ were regular, $L=L_w\cup\{w\}$ would also be regular. Thus, $L_w$ cannot be regular, and each of the infinitely many languages $L_w$ is a distinct non-regular subset of $L$.
