Union of Uncountably Infinite Sets How does one notationally describe the set which is the union of uncountably many other sets.  For instance, for each x such that a < x < b, where a and b are real numbers, if there is assigned a set $N_x$,  how does one describe the union of all $N_x$ for all real x, a < x < b?
 A: If $I$ is any index set, and $N_i$ is a set for $i\in I$ we write: $$\bigcup_{i\in I} N_i$$
Sometimes we write instead: $$\bigcup\{N_i\mid i\in I\}$$
A: Usually something like this:
$$\bigcup_{a<x<b}N_x$$
In general, given any set $\Lambda$ (called "index set" in such a context) along with a collection of sets $A_x$ such that there is $A_x$ for every $x\in\Lambda$, one simply writes
$$\bigcup_{x\in\Lambda} A_x$$
To denote the union. So yo can also use
$$\bigcup_{x\in (a,b)}N_x$$
A: I would write it as $$\bigcup_{a<x<b}{N_x}$$
A: Any of the following is just fine:
$$\bigcup_{x\in(a,b)}N_x = \bigcup_{a<x<b}N_x = \bigcup\{N_x:x\in(a,b)\}=\bigcup\{N_x:a<x<b\}.$$
In general, if you have sets $S_\alpha$ indexed by elements of some set $A$, you can write $$\bigcup_{\alpha\in A}S_\alpha = \bigcup\{S_\alpha:\alpha\in A\}.$$
If you have an unindexed collection $\mathscr{S}$ of sets, you can either write simply $\bigcup\mathscr{S}$ or use the sets themselves as indices: $$\bigcup_{S\in\mathscr{S}}S = \bigcup\{S:S\in\mathscr{S}\}.$$
