Can a countable set of parabolas cover the unit square in the plane? Can a countable set of parabolas cover the unit square in the plane?  My intuition tells me the answer is no, since it can't be covered by countably many horizontal lines (by the uncountability of $[0, 1]$).  Help would be appreciated.

An approach: Let the parabolas be $P_1,P_2,\dots$. Then by thickening each parabola slightly, we can make the area covered by the thickened parabola $P_i^\ast$ less than $\frac{\epsilon}{2^i}$, where $\epsilon$ is any preassigned positive number, say $\epsilon=1/2$.  Then the total area covered by the thickened parabolas is $\lt 1$.  
Remark: There are various ways to say that a set is "small." Cardinality is one such way. Measure is another.  
