Notation for countable union

Question

I sometimes see the following notation for a countable union of sets: $\bigcup_{n\in\omega}E_n$

Is this notation strictly synonymous with the more standard $\bigcup_{n\in\mathbb{N}}E_n$?

Answer 1

$\omega$ is a standard set theory notation for the set $\mathbb{N}$; it's the first infinite ordinal number $\{0,1,2,3,\ldots\}$. With $\mathbb{N}$ you have to be careful whether $0$ belongs to it or not.

So the notations are synonymous. For a set theorist $n \in \omega$ would be more standard; it depends on one's background what one finds standard.