# Definition of Omega Jump [closed]

In the book Computability theory (Rebecca Weber) I stumbled about Exercise 7.1.24 with the definition of the omega jump. The book says: The omega jump of $$A$$, $$A^{(\omega)}$$ is the join of all $$A^{(n)}$$ for $$n \in \mathbb{N}$$: $$A^{(\omega)}=\{\langle x, n \rangle : x \in A^{(n)}\}$$.
I tried to understand this definition but i couldn't. I think I know that the jump is but I dont understand the omega jump. Is it simply $$A^{(\omega)} = \cup_{n \in \mathbb{N}} A^{(n)}$$? Why (not) ?
Can someone give me an equivalent definition or explain it to me?

The union is not a good idea because you cannot tell, if a number is in the union, whether the number got in because it is an element of $$0^{(n)}$$ or $$0^{(n+1)}$$, etc. Instead, if you take the union of the $$$$ where $$a$$ is an element of $$0^{(n)}$$ then you can keep them separate.