Locally non-enumerable dense subsets of R Today after lunch I was hungry for math problems so I started begging for some at the department and finally someone threw me this: Can $\mathbb{R}$ be partitioned into two non-countable dense subsets? It was a good starter, after a few minutes I got: Take the irrationals less than $0$ and the rationals greater than $0$, this is one subset, the complement of course works. Then the following question came into my mind: Can $\mathbb{R}$ be partitioned into two locally-non-countable dense subsets?
I'm still hungry 

Updated in light of the comment:  How about we express all numbers in base 3.  For set $A$ take all the numbers that eventually have $1$'s in all the even places behind the base 3 point-things like $0.x1x1x1x1x1\ldots$ (where the $x$'s can be any digit and need not be the same), $0.xxx1x1x1x1x1x1\ldots$, $0.xxxxx1x1x1x1x1\ldots$ and for $B$ take everything else.
