Definition:Filter on Set

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

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


 * $$X \in \mathcal F$$


 * $$\varnothing \notin \mathcal F$$


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


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

Filtered Set
The set $$X$$ on which the filter has been applied is called a set filtered by $$\mathcal F$$, or just filtered set.

Trivial Filter
A filter $$\mathcal F$$ on a set $$X$$ as defined above specifically does not include the empty set $$\varnothing$$.

If a filter $$\mathcal F$$ were to include $$\varnothing$$, then from Empty Set Subset of All it would follow that every subset of $$X$$ would have to be in $$\mathcal F$$, and so $$\mathcal F = \mathcal P \left({X}\right)$$.

Such a filter is called the trivial filter on $$X$$.

Finite Intersection
It follows directly by Principle of Mathematical Induction from:
 * $$U, V \in \mathcal F \implies U \cap V \in \mathcal F$$

that the intersection of any finite number of sets of $$\mathcal F$$ is also an element of $$\mathcal F$$.

Some treatments of this subject start with this as an axiom.

Also see

 * See Filter on a Set is a Proper Filter.