Definition:Filter on Set

Let $$X$$ be a set, and $$\mathcal{P} \left({X}\right)$$ be the power set of $$X$$.

A filter is a set $$\mathcal{F} \subset \mathcal{P}(X)$$ which satisfies the following conditions:
 * $$\varnothing \not \in \mathcal{F}$$ 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 $$V \in \mathcal{F}$$.