Proposition 2

Let $\mathfrak{B}$ be a set of subsets of a set $X$. Then the set of subsets of $X$ which contain a set of $\mathfrak{B}$ is a filter if and only if $\mathfrak{B}$ has the following two properties: (B_I) The intersection of two sets of $\mathfrak{B}$ contains a set of $\mathfrak{B}$. (B_{II}) $\mathfrak{B}$ is not empty, and the empty subset of $X$ is not in $\mathfrak{B}$.