What do sets in $S := \{ (-\infty, b) : b \in \Bbb{R} \} \cup \{ (a, \infty) : a \in \Bbb{R} \}$ look like If I'm given a collection
$S := \{ (-\infty, b) : b \in \Bbb{R} \} \cup \{ (a, \infty) : a \in \Bbb{R} \}$
Then would the sets of S only be of the form $(-\infty, b) \cup (a, \infty)$ or could they also be just $(-\infty,b) , (a, \infty)$. 
 A: When in doubt, try to work out smaller examples.
If $S$ is the union of $\{\{1\}\}$ and $\{\{2\}\}$. Which one of the three is correct?


*

*$S=\{1,2\}$,

*$S=\{\{1,2\}\}$,

*$S=\{\{1\},\{2\}\}$.


Not sure? Let's try an even simpler example. $S$ is the union of $\{A\}$ and $\{B\}$, what is $S$? If you answered $\{A,B\}$, you're correct. Let's kick it up a notch. $S=\{A_i\mid i\in I\}\cup\{B_j\mid j\in J\}$. What are the elements of $S$?
You should be able to answer your own question now.
A: Well, $S=\{ (-\infty, b) : b \in \Bbb{R} \} \cup \{ (a, \infty) : a \in \Bbb{R} \}$ consists of elements from two sets - since it is the union of two - which is
$$
S_a:=\{ (a, \infty) : a \in \Bbb{R} \}
$$
and
$$
S_b:=\{ (-\infty, b) : b \in \Bbb{R} \} 
$$
which are both uncountable sets of open intervals. If we now take the union $S:=S_a\cup S_b$ this means that we can express $S$ also like
$$
S=\{ (-\infty, b),(a,\infty) : a,b\in \Bbb{R} \} 
$$
with $(-\infty, b)\in S_b$ and $(a,\infty)\in S_a$. So you are actually dealing not with the union of intervals but with the union of two sets which elements are intervals.
