2 Order and algebraic structure

Definition 1 (Filter)

Let $X$ be a set, a filter is a family of subsets of the power set $\mathcal{F}\subseteq \mathcal{P}(X)$ satisfying the following properties:

1. The universal set is in the filter $X\in \mathcal{F}$.
2. If $E\in\mathcal{F}$, then $\forall A\in\mathcal{P}(X)$ such that $E\subseteq A$, we have $A\in\mathcal{F}$.
3. If $E,A\in\mathcal{F}$, then $E\cap A\in\mathcal{F}$.