Have you considered a neighborhood smaller than ½|P-Q|?
I have tried to consider your suggestion, and it looks similar to the epsilon we chose when proving that a sequence can't converge to two numbers unless they were equal. However, I'm not quite sure what I'm supposed to do with that... We just assumed the OP was true and moved on with it.
P.S. I don't have much experience proving things so it's taking me awhile to think things through.
The point is that if a specific works, then so does any positive number less than .
TKHunny suggested which is the largest number that would work so that and are disjoint because . (Draw a picture to see why.) Plato just chose to pick a smaller number. would also work.
And, that is using the same for both sets, which is not necessary. You could also use, say, for one open set and for the other- just use two numbers that add up to less than |P-Q|.