Given a collection $\mathcal A$ of sets, the intersection of elements of $\mathcal A$ is defined by the equation:
$\bigcap_{A\in \mathcal A}A=\{x\mid x\in A \text{ for every } A\in \mathcal A\}.$
If one allows $\mathcal A$ to be 'empty collection' (that is, all the sets in the collection are empty), what then does the definition mean?
The book I'm reading calls the arbitrary intersection in the particular case to be equal to $U$ (the universal set). I'm not quite sure how that can happen.
The author explains how every $x$ vacuously satisfies the given condition but I don't really follow the reasoning he's giving. Help?
Add: Answers received thus far make sense. I'd like to add that the author in the end mentions that not all mathematicians feel it's reasonable to let $\bigcap_{A \in \mathcal A} A = U$ and he too decides to leaves the intersection undefined when $\mathcal A$ is empty in order 'to avoid difficulty.' I wonder why do some have qualms with allowing $U$ to stand for the intersection? Is there another equally reasonable, but contradicting answer to the problem?