Removing countable hyperplanes from a ball in $\mathbb{R}^n$ I would like to claim that an open ball in $\mathbb{R}^n$ cannot be covered by a
countable collection of $(n-1)$-dimensional hyperplanes, that in fact excluding
the hyperplanes from the ball still leaves a set that contains an
uncountable number of points.
For $n=2$, this says removing a countable number of lines from a disk
still leaves an uncountable number of points of the disk.
What is the correct language to justify this?  It's a bit out of my expertise...
Thanks for suggestions!

There are various ways to do this. The most intuitive is probably to show that any hyperplane has zero measure and use the fact that a countable union of sets of zero measure has zero measure. The simplest is probably to take a closed ball inside your open ball and apply the Baire category theorem to it. 
