Definition:Filter on Set

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

A filter is a set $$\mathcal F \subset \mathcal P \left({X}\right)$$ which satisfies the following conditions:


 * $$\varnothing \not \in \mathcal F$$ and $$X \in \mathcal F$$;


 * $$U, V \in \mathcal F \implies U \cap V \in \mathcal F$$;


 * $$U \in \mathcal F: U \subseteq V \subseteq X \implies V \in \mathcal F$$.