Intersection of closed sets is closed

Statement

Let $(X, d)$ be a metric space and let $(F_i)_{i \in I}$ be a family of closed subsets of $X$, indexed by a nonempty set $I$. Then the intersection $\bigcap_{i \in I} F_i$ is closed in $X$.

Remarks

Proof: by De Morgan's laws for sets, $X \setminus \bigcap_i F_i = \bigcup_i (X \setminus F_i)$. This is a union of open sets, so it is open. The same proof works in each topological space. A union of finitely many closed sets is closed, but an infinite union need not be. For example, the union of the sets $[1/n, 1]$ for $n \ge 1$ is $(0, 1]$.

Explanations

L1 human

Closed sets behave like open sets with the roles of union and intersection exchanged. Any intersection of closed sets is closed, and finite unions of closed sets are closed.