Definition:Finite Intersection Axiom

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $T$ be such that:
 * Every set $V_\alpha$ of closed sets of $T$ such that $\displaystyle \bigcap V_\alpha = \varnothing$ contains a finite subset $V_\beta \subseteq V_\alpha$ such that $\displaystyle \bigcap V_\beta = \varnothing$.

Then $T$ satisfies the finite intersection axiom.

Also see

 * Do not confuse this with the Finite Intersection Property.