Q:

Countable infinite intersection of uncountable sets ist countable? 

There is a series of uncountable sets.  Is the the countable infinite intersection of them in every case countable?  And if yes, why? 

A:

No, suppose $A_i=\mathbb{R}$ for any $i\in \mathbb{N}$, then $\bigcap_{i\in \mathbb{N}}A_i=\mathbb{R}$ is uncountable.  It's even worse, finite intersections of uncountable sets may still be uncountable, indeed, $A_1\cap A_2=\mathbb{R}$.

