My textbook, Mathematical Analysis For Machine Learning and Data Mining by Dan Simovici, says the following:
If $\mathcal{C}$ is a non-empty collection, its intersection is the set $\bigcap \mathcal{C}$ given by
$$\bigcap \mathcal{C} = \{x \mid x \in S \ \ \text{for every $S \in \mathcal{C}$} \}$$
Wolfram defines a nonempty set as follows:
A nonempty set is a set containing one or more elements. Any set other than the empty set $\emptyset$ is therefore a nonempty set.
But, given these definitions, would it not be true that the intersection of any non-empty collection is the empty set? After all, the empty set is an element of every set (including itself), which means that, despite the fact that $\mathcal{C}$ itself is a non-empty set (that is, $\mathcal{C} \not= \emptyset$), all of the sets it contains do themselves contain the empty set as an element, and so we would always have that $x = \emptyset$ in the above definition?
I would appreciate it if people would please take the time to clarify this.
