Let $U$ be a set.
Concept = "a set $\mathscr{C}$ of proper filters on $U$ such that if $X\in\mathscr{C}$ and $Y$ is a proper filter on $U$ and $Y\supseteq X$, then $Y\in\mathscr{C}$."
Is there a specific term or a combination of terms to succinctly describe this concept?
If there are no existing term, please help me to devise a good English word to describe this situation (in the book I am writing).