Question

Does there exist a partition of R that is an uncountably infinite partition that consists of uncountably infinite sets? If so, construct such a partition, otherwise prove that such a partition can not exist.

Answer 1

We know that has the same cardinality as . So, there exists a bijection

Now, consider the following family of sets in

where

Then, each and the family forms an uncountably infinite partition of , with each uncountably infinite as well.

Now, consider the image of each of these under the bijection , given by

.

Then, by construction, the family of sets

is the required uncountably infinite partition of that consists of uncountably infinite sets.