- #1
Chasing_Time
- 8
- 0
Hi all, I'm getting stuck on this problem.
I am asked to show that that the open ball in the plane {|x|} < 1} can be written as a countable union of rectangles [a_1, a_2] x [b_1,b_2], but the closed ball in the plane {|x| <= 1} cannot be written as a countable union of rectangles.
I believe it has something to do with the fact that the rationals are countable but the reals are not, and extending this relationship to 2-D. For a given rectangle (a1, a2) X (b1, b2), to get the open ball one could keep halving the distances |a2 - a1| and |b2 - b1| for each rectangle, and since the operation is halving the set the result is still countable, and further, the product will be countable as well as the union of all such rectangles. But to get the closed ball must invoke irrational numbers- I believe this has something to do with the fact that the ball is convex, but I'm not sure how to incorporate that either. At least a starting point would be very helpful. Thanks so much.
Homework Statement
I am asked to show that that the open ball in the plane {|x|} < 1} can be written as a countable union of rectangles [a_1, a_2] x [b_1,b_2], but the closed ball in the plane {|x| <= 1} cannot be written as a countable union of rectangles.
The Attempt at a Solution
I believe it has something to do with the fact that the rationals are countable but the reals are not, and extending this relationship to 2-D. For a given rectangle (a1, a2) X (b1, b2), to get the open ball one could keep halving the distances |a2 - a1| and |b2 - b1| for each rectangle, and since the operation is halving the set the result is still countable, and further, the product will be countable as well as the union of all such rectangles. But to get the closed ball must invoke irrational numbers- I believe this has something to do with the fact that the ball is convex, but I'm not sure how to incorporate that either. At least a starting point would be very helpful. Thanks so much.