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:
 * $$\forall n \in \N: \bigcap_{i = 1}^n S_i \ne \varnothing$$

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

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