Prefilter

Filter bases

This upwards closure is a filter if and only if ${\displaystyle {\mathcal {B}}}$ is downwards-directed, i.e., ${\displaystyle {\mathcal {B}}}$ is non-empty and for all ${\displaystyle A,B\in {\mathcal {B}}}$ there exists ${\displaystyle C\in {\mathcal {B}}}$ such that ${\displaystyle C\subseteq A\cap B}$. When this is the case, ${\displaystyle {\mathcal {B}}}$ is also called a prefilter, and the upwards closure is also equal to the generated filter ${\displaystyle \langle {\mathcal {B}}\rangle }$. Hence, being a filter base of ${\displaystyle {\mathcal {F}}}$ is a stronger property than being a filter subbase of ${\displaystyle {\mathcal {F}}}$.