Regular, CFL, non-CFL infinite closures

Question

I was wondering about infinite closure properties.

Are the Regular languages closed under infinite union? Infinite intersection?

Probably not, by taking $\forall n>0~~L_n=\{a^nb^n\}\in RL$, then $\bigcup_{n=1}^{\infty}L_n=\{a^nb^n|n>0\}\notin RL$

By taking $L_k=\{a^ib^j~|~i\ne k ~~or~~ j\ne k\}\in RL$, then $\bigcap_{k=1}^{\infty}L_k=\{a^ib^j|i \ne j\}\notin RL$

But what about CFL and non-CFL languages? I couldn't find examples for that.

Thanks!