Question

Questions about partitions:

See below where I put in math font.

Answer 1

r >= 0, let C_r = {(x,y) in R^2 | x^2 + y^2 = r^2} and let F = {C_r | r in [0, infinity)}
Is this a partition?

Answer 2

I understand the underlying principles, but I don't understand how to generalize it.

Answer 3

I don't think it is a partition in the usual sense. You have described the set of all circles centered at the origin, without specifying what your values of r are. If you want an infinitely dense partition, let r be an element of Q; i.e., the partitions are at rational numbers. If you need more, there are always more rational numbers between the ones you already have.

Answer 4

After rereading my reply, I guess the term infinitely dense isn't really correct; dense is sufficient.