Definition:Finite Intersection Property

Definition
Let $\mathbb S$ be a set of sets.

Let $S_i \in \mathbb S$ for all $i \in \N$.

Suppose $\mathbb S$ has the property that:
 * $\displaystyle \forall n \in \N: \bigcap_{i = 1}^n S_i \ne \varnothing$

That is, the intersection of any finite number of sets in $\mathbb S$ is not empty.

Then $\mathbb S$ satisfies the finite intersection property.