CFL is closed under countably infinite

Did you mean "CFL is closed under **countably infinite union**"?

Because if you did, I could very easily disprove it.

Take this example.. We know that {$a^nb^nc^n | n \geq 1$} is Non-CFL CSL. And We also know that the language {$a^nb^nc^n | n \geq 1$} is Countably Infinite (Because we know that $\Sigma^*$ is itself countably infinite and every subset of Countably infinite is also Countably Infinite).

Now, Take each string of above language i.e. {$a^nb^nc^n | n \geq 1$} and make into languages of One string Only.

i.e. $\left \{ abc \right \} \cup \left \{ a^2b^2c^2 \right \} \cup \left \{ a^3b^3c^3 \right \} \cup ........$ Here, All the languages are CFL and There are Countably Infinite such languages.... But Union is Not CFL but CSL.

No Family of languages is closed under Infinite Union. Not even NOT RE family.

And Even Set of **Non-Regular CFL** if you consider... Is NOT closed under Infinite Union...It is not even closed under Union..Let alone Infinite union.