Definition:Finite Intersection Property

Definition

Let $\Bbb S$ be a set of sets.

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

Suppose $\Bbb S$ 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 $\Bbb S$ is not empty.

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

Also see

• Do not confuse this with the Finite Intersection Axiom.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Filters