Finite intersection property

A family $\mathcal F$ of subsets of a set $X$ has the finite intersection property if for every finite choice $F_1,\dots,F_n\in\mathcal F$ one has

where $\varnothing$ is the empty set .

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 (any finite intersection is nonempty), but $\bigcap_{n\ge 1}F_n=\varnothing$.
• In a set $X=\{1,2,3,4\}$, the family $\{\{1,2\},\{2,3\},\{2,4\}\}$ has the finite intersection property, since every finite intersection contains $2$.