Why this set is an algebra

Question

My question is, how can I form the empty $\emptyset $ set with finite sets unions as in (9.1)?

Answer 1

There are two ways to make the empty set.

One way is as the interval $(1,0]$, which is empty and of the given form.

The other is as the empty union: $$\bigcup_{I\in\varnothing}I$$This is a finite union, all the sets in the union are of the given form (maybe it's easier to think of as "it doesn't include any sets not of the given form"), and it equals the empty set.