Finite intersection property

A family $\mathcal F$ of subsets of a set $X$ has the finite intersection property if every finite subfamily has nonempty intersection, taking the empty intersection to be $X$ . In particular, for $F_1,\ldots,F_n\in\mathcal F$ ,

F_1\cap\cdots\cap F_n \neq \varnothing,

Compactness application

In topology, the finite intersection property is especially useful for families of closed sets: compactness can be characterized by requiring that every family of closed sets with this property has nonempty total intersection.

Examples

In $\mathbb R$ , the family $F_n=[n,\infty)$ has the finite intersection property, but $\bigcap_{n\ge1}F_n=\varnothing$ .
The family $\{\{1,2\},\{2,3\},\{2,4\}\}$ has the finite intersection property because every finite intersection contains $2$ .