What is the cardinality of the union of uncountable sets

Question

If a set U is the union of all the uncountable sets (open intervals), then what is the cardinality of U? And how to prove that?

Answer 1

If you mean intervals of the reals, the union of all the open intervals, is equal to $\mathbb R$ itself. Therefore this union has the power of the continuum $\frak c=2^{\mathbb N}$ for cardinality.

Answer 2

In your particular question the answer is 2^No You may use the following steps

1. Every interval has cardinality as 2^No
2. Your Uncountable union will be contained in the Space of real numbers
3. Cardinality of the set of Real numbers is 2^No

I hope it helps.