Let's see the definition of a filter with the "$\subset$" order.
Let $X$ be a set. We say a non-empty family $\mathcal{F}$ of subsets of $X$ is a filter if:
- $\emptyset \notin \mathcal{F}$
- if $A, B \in \mathcal{F}$ then $A \cap B \in \mathcal{F}$
- if $A \in \mathcal{F}$ and $A \subset B$, then $B \in \mathcal{F}$
You can't say $$\bigcap_{F \in \mathcal{F}} F \subset \mathcal{F}$$
because not every filter is a principal filter, i.e. the interception above can be the empty set.
A principal filter is a filter generated by a single element.
By filter's statement 2 we know the finite intersection is defined and is an element of the filter. But why can't the interception of all elements of the filter be defined as is the finite intersection?
Maybe my question could be: why there are non-principal filters?