Definition:Filter on Set

Let $$X$$ be a set. A set $$\mathcal{F} \subset \mathcal{P}(X)$$ is called a filter if it satisfies the following conditions:
 * $$\emptyset \not \in \mathcal{F}$$ and $$X \in \mathcal{F}$$
 * If $$U, V \in \mathcal{F}$$ then $$U \cap V \in \mathcal{F}$$
 * If $$U \in \mathcal{F}$$ and $$U \subset V \subset X$$ then $$V \in \mathcal{F}$$