Moore family

Remarks

The closed sets of a topological space $X$ form a Moore family: $X$ is closed, and any intersection of closed sets is closed. The open sets of $\R$ do not: the open intervals $(-1/n, 1/n)$ for $n \ge 1$ have intersection $\set{0}$, which is not open. If $\mathcal{M}$ is a Moore family, then $c(A) = \bigcap \set{M \in \mathcal{M} : A \subseteq M}$ is a closure operator on the subsets of $X$. Its closed sets are exactly the members of $\mathcal{M}$. Conversely, the closed sets of any closure operator on the subsets of $X$ form a Moore family. With the convention that the empty intersection is $X$, the two conditions merge into one: $\mathcal{M}$ is closed under all intersections. Other names are closure system and intersection-closed family.